FLOWS ON S-ARITHMETIC HOMOGENEOUS SPACES AND

APPLICATIONS TO METRIC DIOPHANTINE APPROXIMATION

Dmitry Kleinbock and George Tomanov

Abstract. The goal of this work is to prove quantitative nondivergence estimates for flows on S-arithmetic homogeneous spaces, and then apply them to establish Sarithmetic generalization of conjectures of Baker and Sprindžuk in metric Diophantine approximation. In the present paper we develop the general framework and establish results for polynomial maps. The sequel [KT] will contain applications to Diophantine approximation on differentiable S-arithmetic submanifolds.

0. Introduction

0.1. Actions of unipotent subgroups on homogeneous spaces provide examples of important dynamical systems with numerous applications to geometry and number theory. A fundamental phenomemon discovered by G.A. Margulis in 1971 [M] showed that, in sharp contrast to partially hyperbolic flows, orbits of unipotent flows are never divergent. Margulis’ paper gave rise to numerous generalizations and applications. In a series of papers [D1–3], S.G. Dani generalized and strengthened this result. In particular, for one-parameter flows he proved that any orbit returns to big compact subsets with high frequency. A very general explicit estimate for this frequency in terms of the size of compact sets was given in 1998 in the paper of Margulis and the first named author [KM]. To state one of the results of that paper, let us recall that the homogeneous space SL(m, ℝ)/ SL(m, ℤ) can be interpreted as the space of unimodular lattices in ℝ^{m}, and, in view of Mahler’s Compactness Criterion [Rag, Corollary 10.9], the family of sets

(0.1)  Q_{ε} =^{def} {g SL(m, ℤ) | gℤ^{n} contains no nonzero vector of length < ε}

gives an exhaustion of SL(m, ℝ)/ SL(m, ℤ) by compact subsets. Here is a special case of [KM, Theorem 5.2]:

Theorem. For m, d ∈ ℕ, let h : ℝ^{d} → SL(m, ℝ) be a polynomial map, and fix Λ ∈ SL(m, ℝ)/ SL(m, ℤ). Then there exists positive numbers α (depending only on m, d and the degree of h) and C (depending only on m, d, the degree of h and Λ) such that for any positive ε and any ball B ⊂ ℝ^{d} centered at 0 one has

\frac{λ (\{x ∈ B | h(x)Λ ∉ Q_{ε}\})}{λ(B)} ≤ Cε^{α} .

Typeset by AMS-TEX

(Here λ stands for the Lebesgue measure on ℝ^{d}.)

The results of Margulis and Dani played a crucial role in M. Ratner’s [Rat1–2] proof of conjectures of Raghunathan, Dani and Margulis on orbit closures and invariant measures of groups generated by one-parameter unipotent subgroups. Note that Dani used his results to prove that any locally finite measure invariant under a unipotent flow is finite [D2, Theorem 4.3].

0.2. In [BP], A. Borel and G. Prasad initiated the study of actions of S-adic unipotent groups on homogeneous spaces. The generalization of Ratner’s results to the S-adic setting was achieved in the papers [MT1, Rat3, MT2]. In the present paper we prove the S-adic analogue of explicit quantitative estimates from [KM], and use them to study locally finite unipotent-invariant measures on S-adic homogeneous spaces. Namely, let K be a number field, S a finite set of its normalized valuations containing all the Archimedean ones, O_{S} the ring of S-integers of K, and K_{S} the direct product of completions K_{v} of K over v ∈ S. Put GL(m, K_{S}) =^{def} ∏_{v∈S} GL(m, K_{v}) and

GL^{1}(m, K_{S}) =^{def} {(g^{(v)})_{v∈S} ∈ GL(m, K_{S}) |∏_{v∈S} | det(g^{(v)})|_{v} = 1} .

Then one can interprete the homogeneous space GL^{1}(m, K_{S})/ GL(m, O_{S}) as the space of lattices in K_{S}^{m} of covolume (\sqrt{D_{K}})^{m}, where D_{K} is the discriminant of K, and can define the family of subsets Q_{ε} of GL^{1}(m, K_{S})/ GL(m, O_{S}) in a way similar to (0.1), see §5.12 and §6.4 for details.

Let us now formulate a special case of our Theorem 6.4. Let X = ∏_{v∈S} K_{v}^{d_{v}}, where d_{v} ∈ ℕ, and let h = (h_{v})_{v∈S} : X → GL^{1}(m, K_{S}) be a polynomial map (see §6.4 for the definition) with h(0) = e. We will endow X with the product metric and the measure

(0.2)  λ =^{def} ∏_{v∈S} λ_{v} , where λ_{v} is a Haar measure on K_{v}^{d_{v}} .

Theorem. There exists α > 0 (depending only on m, d_{v} and the degrees of the maps h_{v}) such that for every Λ ∈ GL^{1}(m, K_{S})/ GL(m, O_{S}) one can find C > 0 (depending only on m, d_{v}, the degrees of the maps and Λ) with the following property: for any positive ε and any ball B ⊂ X centered at 0 one has

\frac{λ (\{x ∈ B | h(x)Λ ∉ Q_{ε}\})}{λ(B)} ≤ Cε^{α} .

0.3. Theorem 6.4 makes it possible to generalize the main results from [D1–3] to the case of products of real and p-adic algebraic groups. In particular, here is a special case of our Theorem 7.1:

Theorem. Let G = ∏_{v∈S} G_{v}, where G_{v} is the group of K_{v}-rational points of an algebraic group G_{v} defined over K_{v}, let Γ be a lattice in G, d a natural number, and let X and λ be as in §0.2. Then for any β > 0 there exists a compact M_{β} ⊂ G/Γ such that for any y ∈ G/Γ and any parametrization ϕ : X → U of degree ≤ d of a unipotent algebraic subgroup U of G (see §7.1 for precise definitions) one of the following holds: either

(i) there exists R = R(y, ϕ) such that

\frac{λ (\{x ∈ B | ϕ(x)y ∈ M_{β}\})}{λ(B)} ≥ 1 − β

for any ball B in X centered at 0 with radius at least R, or

(ii) there exists a closed proper subgroup H of G containing U such that the orbit Hy is closed and carries a finite H-invariant Borel measure.

In the particular case when U is one-parameter, Theorem 0.3 has been announced in [MT1, Theorem 11.4], [Rat3, Theorem 9.1] and [T, Theorem 3.3], and used for the proof of Ratner’s uniform distribution theorem [Rat3, Theorem 3] and for other related results.

0.4. The next theorem, which follows from Theorem 7.1, had been proved for real Lie groups by Dani [D1, D2], and for the general case had been announced in [MT1, Theorem 11.5].

Theorem. Let G and Γ be as in Theorem 0.3, let H be a subgroup of G generated by unipotent one-parameter subgroups of G, and let μ be a locally finite H-invariant measure on G/Γ. Then there exist Borel H-invariant subsets Y_{i}, 1 ≤ i < ∞, such that μ(Y_{i}) < ∞ for all i and μ(G/Γ ∖ ⋃_{i} Y_{i}) = 0. In particular, every locally finite H-invariant ergodic measure on G/Γ is finite.

0.5. Another motivation for our work comes from applications to number theory. Indeed, using an S-adic generalization of the correspondence between homogeneous dynamics and Diophantine approximation, we establish S-arithmetic versions of several important results in metric Diophantine approximation over ℝ.

Let us recall several basic facts from the theory of simultaneous real Diophantine approximation. We will consider n-tuples y = (y_{1}, …, y_{n}) of real numbers viewed as linear forms, and will be interested in approximating their values y·q = y_{1}q_{1}+⋯+y_{n}q_{n} at integer points q = (q_{1}, …, q_{n}) by integers. One says that y ∈ ℝ^{n} is very well approximable (VWA) if for some ε > 0 there are infinitely many q ∈ ℤ^{n} such that

(0.3)  |q_{0} + q · y| ≤ ∥q∥^{−n(1+ε)}

for some q_{0} ∈ ℤ, and that it is very well multiplicatively approximable (VWMA) if for some ε > 0 there are infinitely many q ∈ ℤ^{n} such that

|q_{0} + q · y| ≤ Π_{+}(q)^{−(1+ε)}

for some q_{0} ∈ ℤ, where we put Π_{+}(q) =^{def} ∏_{i=1}^{n} max(|q_{i}|, 1).

It is well known and easy to see that the set of VWMA vectors has Lebesgue measure zero and contains the set of VWA vectors. The subject of metric Diophantine approximation with dependent quantities originated with a conjecture of Mahler [M] stating that for almost every x ∈ ℝ, the point

(0.4)  f(x) = (x, x^{2}, …, x^{n})

is not very well approximable. More generally, following [KLW], we will say that a measure μ on ℝ^{n} is extremal (resp. strongly extremal) if μ-a.e. point of ℝ^{n} is not VWA (resp. not VWMA). In these terms, Mahler’s conjecture, proved by Sprindžuk in the 1960s (see [Sp]), states that the pushforward f_{∗}λ of Lebesgue measure λ on ℝ by the map f as in (0.4) is extremal. On the other hand, the strong extremality of f_{∗}λ, conjectured by A. Baker in 1972 [B], has not been proved until the paper [KM], which introduced a dynamical approach to this class of problems.

It will be convenient to use the following terminology: if X is a metric space and μ a measure on X, a map f = (f_{1}, …, f_{n}) from X to F^{n}, where F is a locally compact field, will be called μ-nonplanar at x_{0} ∈ X if for any neighborhood B of x_{0} the restrictions of 1, f_{1}, …, f_{n} to B ∩ supp μ are linearly independent over F; in other words, if f(B ∩ supp μ) is not contained in any proper affine subspace of F^{n}. Note that in the special case when X is F^{d}, λ is Haar measure and f is polynomial, this condition is independent on x_{0}. We will omit the dependence on the measure when it is taken to be Lebesgue or Haar.

Here is a special case of [KM, Theorem A], settling Baker’s conjecture:

Theorem. Let f : ℝ^{d} → ℝ^{n} be a nonplanar polynomial map. Then f_{∗}λ is strongly extremal.

0.6. Since the original work of Sprindžuk, one of the directions of research in metric Diophantine approximation has been studying local fields other than ℝ. In particular, Sprindžuk proved [Sp] that for almost all x ∈ ℚ_{p} and any ε > 0 there are at most finitely many q̃ =^{def} (q_{0}, q_{1}, …, q_{n}) ∈ ℤ^{n+1} such that

(0.5)  |q_{0} + q · y| ≤ ∥q̃∥^{−(n+1)(1+ε)} ,

where y is of the form (0.4). We refer the reader to §8.2 of the present paper for the explanation of why the coefficients n in (0.3) and n + 1 in (0.5) are optimal, and why ∥q∥ in the right hand side of (0.3) has to be replaced by ∥q̃∥. To be consistent, let us say that y ∈ ℚ_{p}^{n} is VWA if for some ε > 0 there are infinitely many q̃ ∈ ℤ^{n+1} such that (0.5) holds, and that it is VWMA if for some ε > 0 there are infinitely many q̃ ∈ ℤ^{n+1} such that

|q_{0} + q · y| ≤ Π_{+}(q̃)^{−(1+ε)} .

As before, say that a measure μ on ℚ_{p}^{n} is extremal (resp., strongly extremal) if μalmost every point of ℚ_{p}^{n} is not VWA (resp. not VWMA). Thus the aforementioned Sprindžuk’s result is the extremality of f_{∗}λ, where f : ℚ_{p} → ℚ_{p}^{n} is of the form (0.4) and λ is a Haar measure on ℚ_{p}; that is, a p-adic analogue of the conjecture of Mahler. It seems natural to ask about a p-adic analogue of Baker’s conjecture, and indeed it has been conjectured by I. Shparlinski [Bernik, private communication] that f_{∗}λ as above is strongly extremal. In this paper we verify this conjecture, and, more generally, prove Theorem. Let f : ℚ_{p}^{d} → ℚ_{p}^{n} be a nonplanar polynomial map. Then f_{∗}λ is strongly extremal.

0.7. In fact, the above theorem is a very special case of much more general Sarithmetic results. We postpone giving precise definitions until §8, where we review an S-adic generalization of real/p-adic metric Diophantine approximation, and in particular define extremality and strongly extremality of a measure on ℚ_{S}^{n} = ∏_{v∈S} ℚ_{v}^{n}. The following can be proved:

Theorem. Let S be a finite set of normailzed valuations of ℚ, let f be of the form (f^{(v)})_{v∈S} : ∏_{v∈S} ℚ_{v}^{d_{v}} → ℚ_{S}^{n}, where each f^{(v)} is a nonplanar polynomial map from ℚ_{v}^{d_{v}} to ℚ_{v}^{n}, and let λ be defined as in (0.2), with K = ℚ. Then f_{∗}λ is strongly extremal.

Note that the paper [Z] considers the case when each f^{(v)} is of the form (0.4), and proves the extremality of f_{∗}λ.

Theorem 0.7 is a special case of Theorem 8.4, which is much more general but requires a certain terminology so we do not state it in the introduction. The advantage of having such a general result is that it is possible to exhibit many non-polynomial maps satisfying the assumptions of Theorem 8.4. This was done in [KM] for so-called smooth nondegenerate maps, and we plan to develop an S-adic version in the sequel [KT] to this paper.

Another advantage is that Theorem 8.4 generalizes the main result of [KLW], which, among other things, studies Diophantine properties of generic points on certain fractal subsets of ℝ^{n}. In fact, following [H], the latter class of sets can be defined in vector spaces over any locally compact field, and Theorem 8.4, combined with certain results of [KLW], implies that natural measures supported on those sets are strongly extremal.

0.8. The structure of the paper is as follows. In §1 and §2 we introduce and discuss the so-called Besicovitch metric spaces, Federer measures and (C, α)-good functions, that is, the language in which our main results are stated. Then in §3 and §4 we generalize a combinatorial estimate of [KM]. In §5 we prove auxiliary results from algebraic number theory important for the S-arithmetic applications. The quantitative S-arithmetic nondivergence is discussed in §6 where, in particular, Theorem 0.2 is proved. §7 is devoted to proving Theorems 0.3 and 0.4. Then in §8 we turn to the S-arithmetic Diophantine approximation, giving all the definitions, stating the most general strong extremality result (Theorem 8.4), and mentioning applications to fractal measures. The proof of Theorem 8.4 breaks into two special cases (when S does or does not contain the Archimedean valuation), which are treated in §9 and §10 respectively. In both cases the argument hinges on a modification of the dynamical approach to real Diophantine approximation as developed in [KM].

Acknowledgements: Part of the work was done during the authors’ collaboration at the University of Lyon and at Brandeis University, and during the second-named author’s stay at Max Planck Institute; the hospitality of these institutions is gratefully acknowledged. The authors are thankful to G.A. Margulis for his interest in this work and valuable comments. Thanks are also due to Vasily Bernik, Yuri Bilu, Elon Lindenstrauss and Barak Weiss for helpful discussions.

§1. Besicovitch covering property

1.1. For a metric space X, x ∈ X and r > 0, we denote by B(x, r) the open ball of radius r centered at x. For a subset B of X and a function f : X → F, where (F, | · |) is a valued field, we let ∥f∥_{B} =^{def} sup_{x∈B} |f(x)|. If μ is a locally finite Borel measure on X and B is a subset of X with μ(B) > 0, we define ∥f∥_{μ,B} to be equal to ∥f∥_{B ∩ supp μ}, which, in case f is continuous and B is open, is the same as the L^{∞}(μ) norm of f|_{B}, i.e.

∥f∥_{μ,B} = sup {c | μ({x ∈ B : |f(x)| > c}) > 0} .

We will say that a metric space X is Besicovitch if there exists a constant N_{X} such that the following holds: for any bounded subset A of X and for any family B of nonempty open balls in X such that

(1.1)  ∀ x ∈ A is a center of some ball of B ,

there is a finite or countable subfamily {B_{i}} of B with

(1.2)  1_{A} ≤ ∑_{i} 1_{B_{i}} ≤ N_{X},

i.e. A ⊂ ⋃_{i} B_{i}, and the multiplicity of that subcovering is at most N_{X}.

Example. Suppose that X is ultrametric, that is, the non-Archimedean triangle inequality dist(x_{1}, x_{2}) ≤ max_{i=1,2} dist(x, x_{i}) holds for all x, x_{1}, x_{2} ∈ X. Then any two balls in X are either disjoint or contain one another (this observation will be repeatedly used throughout the sequel). This implies that any covering of any subset of X by balls has a subcovering of multiplicity 1; thus any separable ultrametric space is Besicovitch with N_{X} = 1 (the separability of X is equivalent to the collection of all its balls being countable).

1.2. Example. The fact that ℝ^{d} is Besicovitch is the content of Besicovitch’s Covering Theorem [Mat, Theorem 2.7] (the statement “N_{ℝ} = 2” is known as Vitali’s Covering Theorem). In fact, Besicovitch’s proof, see [Mat, pp. 29–34], can be easily generalized to give the following

1.3. Lemma. For a metric space X, define

(1.3)  M_{X} =^{def} sup \{k | \begin{matrix}∃ balls B_{i} = B(x_{i}, r_{i}), 1 ≤ i ≤ k, \\ such that ⋂_{i=1}^{k} B_{i} ≠ ∅ , and x_{i} ∉ ⋃_{\begin{matrix}j=1 \\ j≠i \end{matrix}}^{k} B_{j} ∀ i \end{matrix}\} ,

and also, for c > 1,

(1.4)  D_{X}(c) =^{def} sup \{k | \begin{matrix}∃ x ∈ X, r > 0 and pairwise disjoint balls \\ B_{1}, …, B_{k} of radius r contained in B(x, cr) \end{matrix}\} .

Then N_{X} ≤ M_{X}D_{X}(8); hence X is Besicovitch if M_{X} and D_{X}(8) are finite.

Proof. We follow the proof of Theorem 2.6 in [Mat]. Take a bounded A ⊂ X and a family B of nonempty open balls in X satisfying (1.1). For each x ∈ A pick one ball B(x, r(x)) ∈ B. As A is bounded, we may assume that

R_{1} = sup_{x∈A} r(x) < ∞ .

Choose x_{1} ∈ A with r(x_{1}) ≥ R_{1}/2 and then inductively

x_{j+1} ∈ A ∖ ⋃_{i=1}^{j} B(x_{i}, r(x_{i})) with r(x_{j+1}) ≥ R_{1}/2

as long as possible. Since A is bounded, the process terminates and we get a finite sequence x_{1}, …, x_{k_{1}}.

Next let

R_{2} = sup {r(x) |x ∈ A ∖ ⋃_{i=1}^{k_{1}} B(x_{i}, r(x_{i}))} .

Choose x_{k_{1}+1} ∈ A ∖ ⋃_{i=1}^{k_{1}} B(x_{i}, r(x_{i})) with r(x_{k_{1}+1}) ≥ R_{2}/2 and again inductively

x_{j+1} ∈ A ∖ ⋃_{i=1}^{j} B(x_{i}, r(x_{i})) with r(x_{j+1}) ≥ R_{2}/2 .

Continuing this process we obtain an increasing sequence of integers 0 = k_{0} < k_{1} < k_{2} < …, a decreasing sequence of positive numbers R_{i} with 2R_{i+1} ≤ R_{i}, and a sequence of balls B_{i} = B(x_{i}, r(x_{i})) with the following properties. For j ∈ ℕ, let I_{j} = {k_{j−1} + 1, …, k_{j}}. Then one has

(1.4a)  r(x) < R_{j}/2  for j ∈ ℕ, x ∈ A ∖ ⋃_{i=1}^{k_{j}} B_{i} ,
(1.4b)  R_{j}/2 ≤ r(x_{i}) ≤ R_{i}  for i ∈ I_{j} ,
(1.4c)  x_{j+1} ∈ A ∖ ⋃_{i=1}^{j} B_{i}  for j ∈ ℕ ,
(1.4d)  x_{k} ∈ A ∖ ⋃_{l≠j} ⋃_{i∈I_{l}} B_{i}  for j ∈ ℕ, k ∈ I_{j} ,

(1.4e)  A ⊂ ⋃_{i=1}^{∞} B_{i} .

The first three properties follow immediately from the construction.

To prove (1.4d), take j ∈ ℕ, k ∈ I_{j}, l ≠ j and i ∈ I_{l}. If l < j, then x_{k} ∉ B_{i} by (1.4b). If l > j, then x_{i} ∉ B_{k} by (1.4b), and also r(x_{i}) < r(x_{k}) by the construction, hence x_{k} ∉ B_{i}. Finally, to verify (1.4e), observe that since R_{j} → 0 as j → ∞, (1.4a) forces r(x) to be equal to zero for any x ∈ A ∖ ⋃_{i=1}^{∞} B_{i}.

Clearly (1.4e) proves the first inequality in (1.2). To establish the second inequality, assume that a point x ∈ X belongs to k balls B_{i}, say,

x ∈ ⋃_{i=1}^{k} B_{m_{i}} .

Using (1.4d) and (1.3), we see that the indices m_{i} can belong to at most M_{X} different blocks I_{j}. We now claim that for any j ∈ ℕ,

#(I_{j} ∩ {m_{i} | i = 1, …, k}) ≤ D_{X}(8) .

Indeed, fix j ∈ ℕ and write {n_{1}, …, n_{l}} =^{def} I_{j} ∩ {m_{i} | i = 1, …, k}. By (1.4b) and (1.4c), the balls B(x_{n_{i}}, R_{j}/4)), i = 1, …, l, are disjoint and contained in B(x, 2R_{j})), so the claim follows from (1.4). This proves that X is Besicovitch, with N_{X} ≤ M_{X}D_{X}(8). □

1.4. We will use the above lemma to prove that the products of ℝ^{d} and certain ultrametric spaces are Besicovitch. Here and hereafter, the product of two metric spaces (X, dist_{X}) and (Y, dist_{Y}) will always be supplied with the product metric

dist ((x_{1}, y_{1}), (x_{2}, y_{2})) = max ( dist_{X}(x_{1}, x_{2}), dist_{Y} (y_{1}, y_{2})) ,

so that balls in X×Y are products of balls in X and in Y. In particular, this convention forces the product of two ultrametric spaces to be ultrametric as well.

Lemma. If Y is ultrametric, one has M_{X×Y} = M_{X}.

Proof. Assume M_{X×Y} > M_{X}, and choose k > M_{X} and balls B_{i} = B(x_{i}, r_{i}), 1 ≤ i ≤ k, in X × Y such that ⋂_{i=1}^{k} B_{i} ≠ ∅ and x_{i} is not in ⋃_{j=1, j≠i}^{k} B_{j} for each i. Write B_{i} = E_{i} × F_{i}, where E_{i} and F_{i} are projections of B_{i} onto X and Y respectively. Without loss of generality suppose that the sequence {r_{i}} is non-increasing. Since Y is ultrametric and ⋂_{i=1}^{k} F_{i} ≠ ∅, one has F_{k} ⊂ F_{i} for all i. On the other hand, since M_{X} < k, the center of E_{k} mush lie in E_{i} for some i < k, therefore x_{k} mush lie in B_{i}, a contradiction.

The converse inequality is straightforward (and not needed for our purposes). □

1.5. Corollary. If Y is ultrametric, one has

(1.5)  N_{X×Y} ≤ M_{X}D_{X}(8)D_{Y} (8) ;

in particular, X ×Y is Besicovitch if the three constants in the right hand side of (1.5) are finite.

Proof. It suffices to check that D_{X×Y} (8) ≤ D_{X}(8)D_{Y} (8), which is straightforward, as, since Y is ultrametric, any ball in Y of radius 8r is a disjoint union of at most D_{Y} (8) balls of radius r. □

1.6. Example. Let F be a locally compact field with a non-trivial ultrametric valuation, and let p be the number of elements in the residue class field of F, that is, the number of representatives in the closed unit ball (the ring of integers of F) O =^{def} B(0, 1^{+}) modulo the open unit ball (the valuation ideal of F) P =^{def} B(0, 1). Without loss of generality we can, and will from now on, normalize the valuation so that the diameter of P is equal to 1/p. (If F = ℚ_{p}, this way one gets P = ℤ_{p}, P = pℤ_{p}, and | · | = | · |_{p}, the standard p-adic valuation.) Then it is easy to see that for any c ≥ 1, any ball in F of radius cr is a disjoint union of at most p^{[log_{p} c]+1} balls of radius r. Therefore Corollary 1.5, in particular, implies that the metric space

(1.6)  X = ℝ^{d_{0}} × F_{1}^{d_{1}} × ⋯ × F_{l}^{d_{ℓ}}

is Besicovitch for any ultrametric locally compact fields F_{1}, …, F_{ℓ} and any d_{0}, d_{1}, …, d_{ℓ} ∈ ℕ.

1.7. We close the section with a measure-theoretic counterpart of the Besicovitch property. Namely, say that a locally finite Borel measure μ on X is uniformly Federer if there exists D > 0 such that

(1.7)  sup_{r>0} \frac{μ(B(x, 3r))}{μ(B(x, r))} < D for all x ∈ supp μ .

Equivalently, one can replace “3” in (1.7) by any c > 1. In other words, μ is uniformly Federer if and only if for all c > 1 one has

D_{μ}(c) =^{def} sup_{\begin{matrix}x∈supp μ \\ r>0 \end{matrix}} \frac{μ(B(x, cr))}{μ(B(x, r))} < ∞ .

To simplify notation, we are going to write D_{μ} instead of D_{μ}(3). Note that if μ is a uniformly Federer measure on X with supp μ = X, for all c > 1 one automatically has D_{X}(c) ≤ D_{μ}(2c), or even ≤ D_{μ}(c) if X is ultrametric.

It will also be useful to have a non-uniform version of the above definition: following [KLW], we will say that μ as above is Federer^{1} if for μ-a.e. x ∈ X there exists a neighborhood U of x such that μ|_{U} is uniformly Federer.

Example. Let X be as in (1.6), and denote by p_{i} the number of elements in the residue class field of F_{i}, i = 1, …, ℓ. It is clear that any Haar measure λ on X is uniformly Federer, with D_{λ}(c) ≤ c^{d_{0}} ∏_{i=1}^{ℓ}(cp_{i})^{d_{i}}.

§2. (C, α)-good functions

2.1. Let X be a metric space, μ a Borel measure on X, and (F, | · |) a valued field. For a subset U of X and C, α > 0, say that a function f : X → F is (C, α)-good on U with respect to μ if for any open ball B ⊂ U centered in supp μ one has

(2.1)  ∀ ε > 0 μ(\{x ∈ B | |f(x)| < ε\}) ≤ C (\frac{ε}{∥f∥_{μ,B}} )^{α} μ(B) .

1See [S] and [KLW, §6] for a weaker non-uniform version.

Whenever it does not cause confusion, we will omit the reference to the measure μ, and simply say that the functions are (C, α)-good on U. In fact, in all the applications of our results, the metric space X will be the normed ring as in (1.6), and μ will be chosen to be a Haar measure λ on X. In particular, μ will be positive on open sets, so it will be always possible to replace ∥f∥_{μ,B} in (2.1) by ∥f∥_{B} and not pay attention to the restriction of the center of B lying in supp μ. The above definition generalizes the one from [KM], which involved functions on ℝ^{d}, with μ being Lebesgue measure. See however [KLW] where measures supported on proper subsets of ℝ^{d} are considered.

The following properties are immediate from the definition:

Lemma. Let metric spaces X, Y, U ⊂ X, V ⊂ Y and C, α > 0 be given. (a) f is (C, α)-good on U ⇔ so is |f|;

(b) f is (C, α)-good on U ⇒ so is cf ∀ c ∈ F;

(c) f_{i}, i ∈ I, are (C, α)-good on U ⇒ so is f = sup_{i∈I} |f_{i}|;

(d) If f is (C, α)-good on U and c_{1} ≤ \frac{|f(x)|}{|g(x)|} ≤ c_{2} for all x ∈ U, then g is (C(c_{2}/c_{1})^{α}, α)-good on U;

2.2. The next lemma will be useful in dealing with functions on products of metric spaces:

Lemma. Let metric spaces X, Y with measures μ, ν be given. Suppose f is a continuous function on U × V, where U ⊂ X and V ⊂ Y are open subsets, and suppose C, D, α, β are positive constants such that

(2.2a)

for all y ∈ V ∩ supp ν, the function x ↦ f(x, y)

is (C, α)-good on U with respect to μ ,

and

(2.2b)

for all x ∈ U ∩ supp μ, the function y ↦ f(x, y)

is (D, β)-good on V with respect to ν .

Then f is (E, γ)-good on U × V with respect to μ × ν, where

(2.3)  γ = \frac{αβ}{α + β} and E = ((\frac{C}{β} )^{β} (\frac{D}{α} )^{α} (α^{α+β} + β^{α+β}))^{\frac{1}{α+β}} .

Proof. Fix a ball in U × V of the form A × B, where A and B are balls in X and Y, and denote ∥f∥_{μ×ν,A×B} by s. Also denote by φ the function defined on B by

(2.4)  φ(y) =^{def} sup_{x∈A ∩ supp μ} |f(x, y)| ,

and for any η > 0 consider the set

B_{η} =^{def} {y ∈ B | φ(y) ≤ ηs} .

It follows from Lemma 2.1(c) and (2.2b) that φ is (D, β)-good on V with respect to ν, therefore

ν(B_{η}) ≤ D (\frac{ηs}{∥φ∥_{ν,B}} )^{β} ν(B) = Dη^{β}ν(B) .

On the other hand, for y ∈ (B ∩ supp ν) ∖ B_{η} one can write

ν(\{x ∈ A | |f(x, y)| ≤ ε\}) ≤_{(2.2a) and (2.4)} C (\frac{ε}{φ(y)})^{α} μ(A) ≤ C (\frac{ε}{ηs})^{α} μ(A) .

Therefore

\begin{aligned}(μ × ν)(\{(x, y) ∈ A × B | |f(x, y)| ≤ ε\}) & ≤ μ(A)ν(B_{η}) + C (\frac{ε}{ηs})^{α} μ(A)ν(B) \\ & ≤ (C (\frac{ε}{ηs})^{α} + Dη^{β}) μ(A)ν(B) . \end{aligned}

An elementary computation shows that the expression in the right hand side is minimized by choosing η = (Cα/Dβ)^{\frac{1}{α+β}} (ε/s)^{\frac{α}{α+β}}, and the minimum is equal to E (ε/s)^{γ} μ(A)ν(B), with E and γ as in (2.3). □

2.3. Corollary. For j = 1, …, ℓ, let X_{j} be a metric space, μ_{j} a measure on X_{j}, U_{j} ⊂ X_{j} open, f_{j} a function which is (C_{j}, α_{j})-good on U_{j} with respect to μ_{j}. Then f : (x_{1}, …, x_{ℓ}) ↦ f_{1}(x_{1}) · . . . · f_{ℓ}(x_{ℓ}) is (C, α)-good on U_{1} × ⋯ × U_{ℓ} with respect to μ_{1} × ⋯ × μ_{ℓ}, with C and α explicitly computed in terms of C_{j}, α_{j}.

2.4. The papers [KM] and [BKM] describe various classes of real valued functions on open subsets of ℝ^{d} which are (C, α)-good with respect to Lebesgue measure. For example, the fact that polynomials in one real variable of degree ≤ k have that property (with α = 1/k) follows easily from Lagrange’s interpolation, see [DM, Lemma 4.1] and [KM, Lemma 3.2]. Similarly, following [T], one can consider polynomials over ultrametric fields:

Lemma. Let F be a locally compact ultrametric valued field. Then for any d, k ∈ ℕ, any polynomial f ∈ F [x_{1}, …, x_{d}] of degree not greater than k is (C_{d,k}, 1/dk)-good on F^{d} with respect to λ, where C_{d,k} is a constant depending only on d and k.

Proof. The case k = 1 is proved in [T, Lemma 4.1], and the general case can be easily derived from the case k = 1 and Lemma 2.2 by induction on d. □

2.5. We will need the following terminology: if (F, | · |) is a valued field, X a metric space, μ a measure on X and f a map from X to F^{n}, we will say that f is μ-good at x_{0} ∈ X (cf. [K2]) if there exists a neighborhood V of x_{0} and positive C, α such that any linear combination of 1, f_{1}, …, f_{n} is (C, α)-good on V with respect to μ. Again, the reference to the measure will be omitted when μ = λ. For example, it follows from the aforementioned results that polynomial maps are good at every point with respect to Lebesgue measure.

It has also been shown in [KM] that the (C, α)-good property can be established for differentiable ℝ-valued functions on subsets of ℝ^{d} with non-vanishing derivatives of some order. In particular, it follows from the results of [KM, §3] that for any function f which is real analytic near x_{0} ∈ ℝ^{d} there exist C, α and a neighborhood U of x_{0} such that f is (C, α)-good on U with respect to Lebesgue measure; consequently, any real analytic map ℝ^{d} → ℝ^{n} is good.

It is possible to prove analogues of the aforementioned results for certain appropriately defined classes of differentiable functions (and in particular for analytic ones) on F^{d}, where F is a locally compact ultrametric field. This produces numerous non-Archimedean examples of (C, α)-good functions and maps. More detail will appear in [KT].

§3. Maps of posets into spaces of good functions

3.1. The goal of this section is to generalize a construction described in [KM] in order to make it work for functions defined on arbitrary metric spaces^{2}. More precisely, we will work with mappings of partially ordered sets (posets) into spaces of functions on a metric space X with a measure μ. Given such a mapping, we will mark certain points (see the definition below), and prove an upper estimate (Theorem 3.1) for the measure of the set of “unmarked” points.

For a poset 𝔓, we will denote by l(𝔓) the length of 𝔓, i.e. the number of elements in a maximal linearly ordered subset of 𝔓. If 𝔖 is a subset of 𝔓, we let 𝔓(𝔖) be the poset of elements of 𝔓 ∖ 𝔖 comparable with any element of 𝔖. Note that one always has

(3.1)  l(𝔓(𝔖)) ≤ l(𝔓) − l(𝔖) .

We will fix a metric space X, and consider posets 𝔓 together with a mapping ψ from 𝔓 to the space C(B) of ℝ-valued continuous functions on some subset B of X, to be denoted by s ↦ ψ_{s}. Given such a mapping and positive numbers ε ≤ ρ, we will say that a point z ∈ B is (ε, ρ)-marked relative to 𝔓 if there exists a linearly ordered subset 𝔖_{z} of 𝔓 such that (M1) ε ≤ |ψ_{s}(z)| ≤ ρ ∀ s ∈ 𝔖_{z}; (M2) |ψ_{s}(z)| ≥ ρ ∀ s ∈ 𝔓(𝔖_{z}). We will denote the set of all such points by Φ(ε, ρ, 𝔓). When it does not cause confusion, we will omit the reference to either 𝔓 or (ε, ρ), and will simply say that z is (ε, ρ)-marked, or marked relative to 𝔓.

Theorem. Let X be a Besicovitch metric space, μ a uniformly Federer measure on X, m ∈ ℤ_{+} and C, α, ρ > 0. Suppose that we are given a poset 𝔓, a ball B = B(x, r) in X, and a mapping ψ : 𝔓 → C(B̃), where B̃ =^{def} B(x, 3^{m}r), such that the following holds: (A0) l(𝔓) ≤ m; (A1) ∀ s ∈ 𝔓 , ψ_{s} is (C, α)-good on B̃ with respect to μ; (A2) ∀ s ∈ 𝔓 , ∥ψ_{s}∥_{μ,B} ≥ ρ; (A3) ∀ y ∈ B̃ ∩ supp μ, #{s ∈ 𝔓 | |ψ_{s}(y)| < ρ} < ∞. Then ∀ ε ≤ ρ one has

2A possibility of such a generalization is mentioned in [KM, §6.1]. The paper [KLW] contains a slightly different presentation of the same argument, written in the special case of 𝔓 being the poset of nonzero rational subspaces of ℝ^{m}.

μ(B ∖ Φ(ε, ρ, 𝔓)) ≤ mC(N_{X}D_{μ}^{2})^{m} (\frac{ε}{ρ})^{α} μ(B) .

Proof. We proceed by induction on m. If m = 0, the poset 𝔓 is empty, and for any z ∈ B one can take 𝔖_{z} = ∅ and check that (M1) and (M2) are satisfied for all ε, ρ; thus all points of B are marked. Now take m ≥ 1 and suppose that the claim is proved for all smaller values of m.

Fix C, α, ρ, b, 𝔓, B = B(x, r) and ψ as in the formulation of the theorem. For any y ∈ B ∩ supp μ define

H(y) =^{def} {s ∈ 𝔓 | |ψ_{s}(y)| < ρ};

this is a finite subset of 𝔓 in view of (A3). If H(y) is empty, y is clearly (ε, ρ)-marked for any positive ε: indeed, since |ψ_{s}(y)| ≥ ρ for all s ∈ 𝔓, one can again take 𝔖_{y} to be the empty set and check that (M1) and (M2) are satisfied. Thus one only needs to consider points y from the set

E =^{def} {y ∈ B ∩ supp μ | H(y) ≠ ∅} = {y ∈ B ∩ supp μ | ∃ s ∈ 𝔓 with |ψ_{s}(y)| < ρ} .

Take y ∈ E and s ∈ H(y), and define

(3.2)  r_{s,y} =^{def} sup{t > 0 | ∥ψ_{s}∥_{μ,B(y,t)} < ρ} .

It follows from the continuity of functions ψ_{s} that for small enough positive t one has ∥ψ_{s}∥_{μ,B(y,t)} < ρ, hence r_{s,y} > 0. Denote B(y, r_{s,y}) by B_{s,y}. From (A1) it is clear that B_{s,y} does not contain B; therefore one has r_{s,y} < 2r. Note also that (3.2) immediately implies that

(3.3)  ∥ψ_{s}∥_{μ,B_{s,y}} ≤ ρ .

Now for any y ∈ E choose an element s_{y} of H(y) such that r_{s_{y},y} ≥ r_{s,y} for all s ∈ H(y) (this can be done since H(y) is finite). For brevity let us denote r_{s_{y},y} by r_{y} and B_{s_{y},y} = ⋃_{s∈H(y)} B_{s,y} by B_{y}. Also let us denote the poset 𝔓({s_{y}}) by 𝔓_{y}.

3.1.1. The next lemma allows one to show a point z ∈ B_{y} to be marked relative to 𝔓 once it is marked relative to 𝔓_{y}. It is proved by a verbatim repetition of the proof of [KM, Lemma 4.6], yet we do it in full detail here to make the argument self-contained.

Lemma. For ε ≤ ρ and y ∈ E, let z ∈ B_{y}∩supp μ∩Φ(ε, ρ, 𝔓_{y}) be such that |ψ_{s_{y}}(z)| ≥ ε; then z belongs to Φ(ε, ρ, 𝔓). Equivalently,

(3.4) (B_{y} ∩ supp μ) ∖ Φ(ε, ρ, 𝔓) ⊂ (B_{y} ∖ Φ(ε, ρ, 𝔓_{y})) ∪ {z ∈ B_{y} | |ψ_{s_{y}}(z)| < ε} .

Proof. By definition of Φ(ε, ρ, 𝔓_{y}), there exists a linearly ordered subset 𝔖_{y,z} of 𝔓_{y} such that

(3.5)  ε ≤ |ψ_{s}(z)| ≤ ρ ∀ s ∈ 𝔖_{y,z}

and

(3.6)  |ψ_{s}(z)| ≥ ρ ∀ s ∈ 𝔓_{y}(𝔖_{y,z}) .

Put 𝔖_{z} =^{def} 𝔖_{y,z} ∪ {s_{y}}. Then 𝔓(𝔖_{z}) = 𝔓_{y}(𝔖_{y,z}); therefore (M2) immediately follows from (3.6), and, in view of (3.5), it remains to check (M1) for s = s_{y}. The latter is straightforward: |ψ_{s_{y}}(z)| is not less than ε by the assumption and is not greater than ρ in view of (3.3). □

3.1.2. Note that one clearly has r_{y} < 2r, which in particular implies that B_{y} ⊂ B(x, 3r). We are going to fix some r′_{y} strictly between r_{y} and min(2r, 3r_{y}), and denote B(y, r′_{y}) by B′_{y}. Clearly one has

(3.7)  ∥ψ_{s}∥_{μ,B′_{y}} ≥ ρ for any y ∈ E and s ∈ 𝔓 .

(Indeed, the definition of r_{y} and (3.2) imply the above inequality for any s ∈ H(y), and it obviously holds if s ∉ H(y).)

Now observe that 𝔓_{y}, B′_{y} and B̃′_{y} =^{def} B(y, 3^{m−1}r′_{y}) satisfy properties

• (A0) with m replaced by m − 1  —  in view of (3.1);
• (A2)  —  in view of (3.7);
• (A1) and (A3)  —  since

B̃′_{y} = B(y, 3^{m−1}r′_{y}) ⊂ B(x, 3^{m−1}r′_{y} + r) ⊂ B(x, (2 · 3^{m−1} + 1)r) ⊂ B(x, 3^{m}r) = B̃ .

Therefore one has

(3.8)  \begin{aligned}μ(B_{y} ∖ Φ(ε, ρ, 𝔓_{y})) & ≤ μ(B′_{y} ∖ Φ(ε, ρ, 𝔓_{y})) \\ & ≤ (m − 1)C(N_{X}D_{μ}^{2})^{m−1} (\frac{ε}{ρ})^{α} μ(B′_{y}) \\ & ≤ D_{μ}(m − 1)C(N_{X}D_{μ}^{2})^{m−1} (\frac{ε}{ρ})^{α} μ(B_{y}) \end{aligned}

by the induction assumption and the Federer property of μ. On the other hand, in view of ψ_{s_{y}} being (C, α)-good on B̃ ⊃ B′_{y}, one can write

(3.9) \begin{aligned}μ (\{z ∈ B_{y} | |ψ_{s_{y}}(z)| < ε\}) & ≤ μ (\{z ∈ B′_{y} | |ψ_{s_{y}}(z)| < ε\}) \\ & ≤ C (\frac{ε}{∥ψ_{s_{y}}∥_{μ,B′_{y}}} )^{α} μ(B′_{y}) ≤_{(3.7)} C (\frac{ε}{ρ})^{α} μ(B′_{y}) \\ & ≤ CD_{μ} (\frac{ε}{ρ})^{α} μ(B_{y}) . \end{aligned}

14 Recall that we need to estimate the measure of E ∖ Φ(ε, ρ, 𝔓). For any y ∈ E, in view of (3.4), (3.8) and (3.9) one has

(3.10)  \begin{aligned}μ(B_{y} ∖ Φ(ε, ρ, 𝔓)) & ≤ C ((m − 1)N m−1_{X}D_{μ}^{2m−1} + D_{μ}) (\frac{ε}{ρ})^{α} μ(B_{y}) \\ & ≤ mCN m−1_{X}D_{μ}^{2m−1} (\frac{ε}{ρ})^{α} μ(B_{y}) . \end{aligned}

Now consider the covering {B_{y} | y ∈ E} of E, choose a countable subset Y of E such that the multiplicity of the subcovering {B_{y} | y ∈ Y } is at most N_{X}, and write

(3.11)  ∑_{y∈Y} μ(B_{y}) ≤ N_{X}μ( ⋃_{y∈Y} B_{y}) ≤ N_{X}μ(B(x, 3r)) ≤ N_{X}D_{μ}μ(B) .

Therefore the measure of E ∖ Φ(ε, ρ, 𝔓) is bounded from above by

\begin{aligned}∑_{y∈Y} μ(B_{y} ∖ Φ(ε, ρ, 𝔓)) & ≤_{(3.10)} mCN m−1_{X}D_{μ}^{2m−1} (\frac{ε}{ρ})^{α} ∑_{y∈Y} μ(B_{y}) \\ & ≤_{(3.11)} mC(N_{X}D_{μ}^{2})^{m} (\frac{ε}{ρ})^{α} μ(B) . □ \end{aligned}

4. Primitive submodules of D^{m}

4.1. We start this section by assuming that

• D is an integral domain, that is, a commutative ring with 1 and without zero divisors;

• K is the quotient field of D;

• R is a commutative ring containing K as a subring. We need the following elementary lemma:

Lemma. Let k, m ∈ ℕ, k ≤ m, and let γ_{1}, …, γ_{k} ∈ K^{m} be linearly independent over K. Then they are linearly independent over R.

Proof. Let A be the m × k-matrix with columns given by γ_{1}, …, γ_{k}. Then there exists at least one k × k minor B of A with det(B) being a nonzero element of K, hence invertible in R. By Cramer’s rule, for any solution β = (β_{1}, …, β_{k}) ∈ R^{k} of Aβ = 0 one must have det(B)β_{i} = 0 for every i, hence β = 0. □

4.2. If Δ is an D-submodule of R^{m}, let us denote by KΔ (resp. RΔ) its K- (resp. R-) linear span inside R^{m}, and define the rank rk Δ of Δ by

(4.1)  rk(Δ) =^{def} dim_{K}(KΔ) .

For example, one has rk(D^{m}) = m for any m ∈ ℕ. If Λ is an D-submodule of R^{m} and Δ is a submodule of Λ, say that Δ is primitive in Λ if any submodule of Λ of rank equal to rk(Δ) and containing Δ is equal to Δ. It is clear that the set of nonzero primitive submodules of a fixed D-submodule Λ of R^{m} is a partially ordered set (with respect to inclusion) of length equal to rk(Λ).

The next lemma characterizes primitive submodules of D^{m}:

Lemma. The following are equivalent for a submodule Δ of D^{m}:

(i) Δ is primitive;

(ii) Δ = KΔ ∩ D^{m};

(iii) Δ = RΔ ∩ D^{m} for any commutative ring R containing K as a subring.

Proof. If Δ = {0}, the claim is trivial. Otherwise, it is obvious that (iii)⇒(ii)⇒(i). Assuming (i) and taking γ ∈ RΔ ∩ D^{m}, let γ_{1}, …, γ_{k} ∈ Δ be a basis of KΔ, with k = rk(Δ). Then γ, γ_{1}, …, γ_{k} are linearly dependent over R, hence, in view of Lemma 4.1, over K. But γ_{1}, …, γ_{k} are linearly independent over K, thus γ belongs to KΔ, therefore the D-module Δ′ generated by Δ and γ has rank k. By primitivity of Δ, Δ′ = Δ, i.e. γ ∈ Δ. □

In fact, Lemma 4.2 implies that for any Δ′ ⊂ D^{m} there exists a unique primitive Δ ⊃ Δ′ of the same rank, namely, Δ = KΔ′ ∩ D^{m}.

4.3. Remark. Note that if D is not a principal ideal domain, then there exist primitive submodules which are not free. For example, let Δ′ = Da, where a = (a_{1}, ..., a_{m}) ∈ D^{m}. Then the primitive submodule Δ = KΔ′ ∩ D^{m} is free if and only if the ideal I in D generated by a_{1}, ..., a_{m} is principal. Indeed, take x ∈ K. Then xa ∈ Δ ⟺ xa_{i} ∈ D for all i ⟺ x ∈ I^{−1}. Therefore Δ = I^{−1}a. But I^{−1} ≅ Δ as D-modules. Hence Δ is free ⟺ I^{−1} is principal ⟺ I is principal, which proves the claim.

Let us now assume in addition that R is a topological ring, and consider the topological group GL(m, R) of automorphisms of R^{m}, i.e. the group of m × m invertible matrices with entries in R. Any g ∈ GL(m, R) maps D-submodules of R^{m} to Dsubmodules of R^{m}, preserving their rank and the inclusion relation. Let us introduce the following notation

(4.2)  𝔐(R, D, m) =^{def} {gΔ | g ∈ GL(m, R), Δ is a submodule of D^{m}} ,

and 𝔓(D, m) =^{def} the set of all nonzero primitive submodules of D^{m} .

Note that the inclusion relation makes 𝔓(D, m) a poset of length m.

4.4. We would like to have a way to measure “size” of submodules from the above collection. Specifically, let us say that a function ν : 𝔐(R, D, m) ↦ ℝ_{+} is norm-like if the following three properties hold: (N1) For any Δ, Δ′ ∈ 𝔐(R, D, m) with Δ′ ⊂ Δ and rk(Δ′) = rk(Δ) one has

ν(Δ′) ≥ ν(Δ); (N2) there exists C_{ν} > 0 such that for any Δ ∈ 𝔐(R, D, m) and any γ ∉ RΔ one

has ν(Δ + Dγ) ≤ C_{ν} · ν(Δ)ν(Dγ); (N3) for every submodule Δ of D^{m}, the function GL(m, R) → ℝ_{+}, g ↦ ν(gΔ), is

continuous. If ν is as above and γ ∈ R^{m}, we will define ν(γ) to be equal to ν(Dγ). The

model example is given by taking D = ℤ, K = ℚ, R = ℝ. Then the set 𝔐(R, D, m) coincides with the set of all discrete subgroups of ℝ^{n}, and one can define ν(Δ) to be the covolume of Δ in ℝΔ, with ν(v) being equal to the Euclidean norm of a vector v ∈ ℝ^{m}; in that case one can easily check that (N1)–(N3) are satisfied, with C_{ν} = 1. In the next section we will do this in a more general context, when R is not a field anymore.

Now we can apply Theorem 3.1 to the poset 𝔓(D, m).

Theorem. Let X be a Besicovitch metric space, μ a uniformly Federer measure on X, and let D ⊂ K ⊂ R be as above, R being a topological ring. For m ∈ ℕ, let a ball B = B(x_{0}, r_{0}) ⊂ X and a continuous map h : B̃ → GL(m, R) be given, where B̃ stands for B(x_{0}, 3^{m}r_{0}). Also let ν be a norm-like function on 𝔐(R, D, m). For any Δ ∈ 𝔓(D, m) denote by ψ_{Δ} the function x ↦ ν(h(x)Δ) on B̃. Now suppose for some C, α > 0 and 0 < ρ < 1/C_{ν} one has

(i) for every Δ ∈ 𝔓(D, m), the function ψ_{Δ} is (C, α)-good on B̃ with respect to μ; (ii) for every Δ ∈ 𝔓(D, m), ∥ψ_{Δ}∥_{μ,B} ≥ ρ; (iii) ∀ x ∈ B̃ ∩ supp μ, #{Δ ∈ 𝔓(D, m) | ψ_{Δ}(x) < ρ} < ∞.

Then for any positive ε ≤ ρ one has

(4.3)  μ (\{x ∈ B | \begin{matrix}ν(h(x)γ) < ε for \\ some γ ∈ D^{m} ∖ \{0\} \end{matrix}\}) ≤ mC(N_{X}D_{μ}^{2})^{m} (\frac{ε}{ρ})^{α} μ(B) .

Proof. For simplicity let us denote 𝔓(D, m) by 𝔓. As was observed above, the length of 𝔓 is equal to m, and one immediately verifies that conditions (i)–(iii) imply that 𝔓, B and B̃ satisfy properties (A1)–(A3) of Theorem 3.1. Thus it suffices to prove that for any positive ε ≤ ρ one has

(4.4)  Φ(ε, ρ, 𝔓) ⊂ {x ∈ B | ν(h(x)γ) ≥ ε for all γ ∈ D^{m} ∖ {0}} .

Take an (ε, ρ)-marked point x ∈ B, and let {0} = Δ_{0} ⊊ Δ_{1} ⊊ ⋯ ⊊ Δ_{l} = D^{m} be all the elements of 𝔖_{x} ∪ {{0}, D^{m}}. Pick any γ ∈ D^{m} ∖ {0}. Then there exists i, 1 ≤ i ≤ l, such that γ ∈ Δ_{i} ∖ Δ_{i−1}. From the primitivity of Δ_{i−1} and Lemma 4.2 it follows that γ ∉ RΔ_{i−1}, hence gγ ∉ gRΔ_{i−1} = RgΔ_{i−1} for any g ∈ GL(m, R). Therefore, if one defines Δ′ =^{def} DΔ_{i−1} + Dγ, in view of (N2) one has

(4.5)  ν(h(x)Δ′) ≤ C_{ν}ν(h(x)Δ_{i−1})ν(h(x)γ) .

Further, let Δ =^{def} KΔ′ ∩ D^{m}. It is a primitive submodule containing Δ′ and of rank equal to rk(Δ′), so, by (N1),

(4.6)  ν(h(x)Δ) ≤ ν(h(x)Δ′) .

Moreover, it is also contained in Δ_{i}, since

Δ = KΔ′ ∩ D^{m} = KΔ ∩ D^{m} ⊂ KΔ_{i} ∩ D^{m} = Δ_{i} .

Therefore it is comparable to any element of 𝔖_{x}, i.e. belongs to 𝔖_{x} ∪ 𝔓(𝔖_{x}). Then one can use properties (M1) and (M2) to deduce that

|ψ_{Δ}(x)| = ν(h(x)Δ) ≥ min(ε, ρ) = ε ,

and then, in view of (4.5) and (4.6), conclude that

ν(h(x)γ) ≥ ν(h(x)Δ)/C_{ν}ν(h(x)Δ_{i−1}) ≥ ε/C_{ν}ρ ≥ ε .

This shows (4.4) and completes the proof of the theorem. □

5. Discrete submodules of K_{S}^{m}

5.1. The goal of this section is to describe a certain class of triples D ⊂ K ⊂ R and construct a norm-like function on 𝔐(R, D, m) which is important in applications to both dynamics and Diophantine approximation.

We take K to be a number field, that is, a finite extension of ℚ, and let S be a finite set of places of K containing all the Archimedean ones. We recall that every completion K_{v} of K is isomorphic to a finite extension of either ℝ or ℚ_{p}. We wil denote by | · |_{v} the normalized valuation on K_{v}. In other words, | · |_{v} is the usual absolute value if v is real, the square of the absolute value if v is complex, and is defined as in Example 1.6 if v is ultrametric.

We let K_{S} be the direct product of all the completions K_{v}, v ∈ S, in which K is diagonally imbedded. Let ∞ ⊂ S stand for the subset of all Archimedean valuations, and let us denote S ∖ ∞ by S_{f}. Also let

K_{∞} =^{def} ∏_{v∈∞} K_{v}, O_{f} =^{def} ∏_{v∈S_{f}} O_{v} and K_{f} =^{def} K_{∞} × O_{f} ,

where for every v ∈ S we denote by O_{v} the ring of integers of K_{v}. We also let O stand for the ring of integers of K, and denote by

O_{S} =^{def} {x ∈ K_{S} | |x^{(v)}|_{v} ≤ 1 for all v ∉ S}

the ring of S-integers of K.

Denote by λ_{v} the normalized (see [W, I.V.4]) Haar measure on K_{v}, and by λ_{S} = ∏_{v∈S} λ_{v} the product measure on K_{S}. Elements of K_{S} will be denoted as x = (x^{(v)})_{v∈S} or simply x = (x^{(v)}), where x^{(v)} ∈ K_{v}. For x of this form, we define the content c(x) of x to be the product of all |x^{(v)}|_{v}, v ∈ S. Since all the valuations are normalized, one has

(5.1)  λ_{S}(xM) = c(x)λ_{S}(M) .

where x ∈ K_{S} and M is a measurable subset of K_{S}.

If m is a natural number, we preserve the same notations λ_{v} and λ_{S} to denote the product measure on K_{v}^{m} and K_{S}^{m}, respectively. Elements x = (x_{1}, …, x_{m}) of K_{S}^{m} will be denoted as x = (x^{(v)}), where x^{(v)} = (x_{1}^{(v)}, …, x_{m}^{(v)}) ∈ K_{v}^{m}. We denote by ∥ · ∥_{v} the square of the usual (Hermitian) norm on K_{v}^{m} if v is complex, the usual (Euclidean) norm on K_{v}^{m} if v is real, and the l^{∞} norm defined by

∥(x_{1}^{(v)}, …, x_{m}^{(v)})∥_{v} = max_{i} |x_{i}^{(v)}|_{v}

if v is non-Archimedean. For x = (x^{(v)}) in K_{S}^{m} we define the content c(x) of x as the product of all the numbers ∥(x^{(v)})∥_{v}, v ∈ S. The group GL(m, K_{S}) = ∏_{v∈S} GL(m, K_{v}) acts naturally on K_{S}^{m}, and one has

λ_{S}(gM) = c( det(g))λ_{S}(M),

where M ⊂ K_{S}^{m} is any measurable subset of K_{S}^{m}, g = (g^{(v)}) ∈ GL(m, K_{S}), and det(g) = ( det(g^{(v)})) ∈ K_{S}^{∗}.

5.2. Our next goal is to define the normalized Haar measure on free K_{S}-submodules of K_{S}^{m}. Let L be a free K_{S}-module of rank r generated by a_{1}, …, a_{r} ∈ K_{S}^{m}. Then one can write L = ∏_{v∈S} L_{v}, where L_{v} = K_{v}a_{1}^{(v)} ⊕ ⋯ ⊕ K_{v}a_{r}^{(v)}. Let us fix a basis (b_{1}, …, b_{r}) of L with the following properties:

(5.2a)  if v ∈ S_{f}, then (b_{1}^{(v)}, …, b_{r}^{(v)}) is an orthonormal basis of L_{v};

(5.2b)  if v ∈ ∞, then L_{v} ∩ O_{v}^{m} = O_{v}b_{1}^{(v)} + ⋯ + O_{v}b_{r}^{(v)} .

Then consider the K_{S}-linear map sending the standard basis (e_{1}, …, e_{r}) of K_{S}^{r} to (b_{1}, …, b_{r}), and define the volume on L as the pushforward of λ_{S} (the normalized Haar measure on K_{S}^{r}) by this map. When it does not lead to confusion, we will denote this measure on L by λ_{S} as well.

Note that the existence of the above basis in the case (5.2b) easily follows from the fact that the ring O_{v} is a principal ideal domain. Note also that the above definition does not depend on the choice of the basis (b_{1}, …, b_{r}) satisfying (5.2ab), because if (b′_{1}, …, b′_{r}) is another such basis and h is an isomorphism of L sending the first one to the second one, then c( det(h)) = 1, which implies that h is measure-preserving.

For any r = 1, …, m we will also consider the r-th exterior power

⋀^{r} K_{S}^{m} ≃ ⨁_{v∈S} ⋀^{r} K_{v}^{m}

of K_{S}^{m}, which is a free K_{S}-module with the standard basis

{e_{i_{1}} ∧ ⋯ ∧ e_{i_{r}} | 1 ≤ i_{1} < ⋯ < i_{r} ≤ m} ,

where e_{1}, …, e_{m} is the standard basis of K_{S}^{m}. We will keep the notations λ_{v}, λ_{S}, ∥·∥_{v} and c(·) to denote the measures, the norms and the content on the exterior powers ⋀^{r} K_{v}^{m} and ⋀^{r} K_{S}^{m}, respectively.

5.3. Recall (cf. [W]) that O_{S} is a lattice in K_{S} with covolume \sqrt{D_{K}}, where D_{K} is the discriminant of K. Any finitely generated O_{S}-submodule of K is called a fractional ideal of K (with respect to O_{S}). The (fractional) ideal I is called an integer ideal (respectively, a principal ideal) if I ⊂ O_{S} (respectively, I = O_{S}α for some α ∈ K). Recall that the set P_{S}(K) (respectively, P_{S}(K)^{0}) of all non-zero fractional (respectively, principal) ideals is an abelian group with respect to multiplication of fractional ideals in K, with O_{S} being the neutral element of this group. The group P_{S}(K) is generated by the principal ideals of O_{S}. The quotient group Cl_{S}(K) = P_{S}(K)/P_{S}(K)^{0} is finite and is called the group of ideal classes. If I is a non-zero integer ideal, then I is a subgroup of finite index of the abelian group O_{S}; this index is called the norm of I and is denoted by N(I). The map N(·) can be extended to a group homomorphism N : P_{S}(K) → ℚ^{+}. (The above well-known facts are proved in [W] in the case when S consists of Archimedean valuations, and a generalization for any S can be easily deduced from this particular case.)

Since O_{S} is a Dedekind domain, in view of [CR, 22.5] we have the following Proposition. Let Δ be a finitely generated torsion-free module over O_{S} of rank r (defined as in (4.1)). Then there exist fractional ideals I_{1}, …, I_{r} such that Δ ≅ I_{1} ⊕ ⋯ ⊕ I_{r}.

We also have

5.4. Proposition. Let Δ be a discrete O_{S}-submodule of K_{S}^{m} of rank r. Then it is finitely generated, torsion-free, and there exist linear independent over K elements a_{1}, …, a_{r} ∈ K_{S}^{m} and integer ideals I_{1}, …, I_{r} such that

(5.3)  Δ = I_{1}a_{1} ⊕ ⋯ ⊕ I_{r}a_{r} .

Furthermore, there exists g ∈ GL(m, K_{S}) such that Δ is contained in gO_{S}^{m}.

Proof. Let Δ be a discrete O_{S}-submodule, and denote the intersection of Δ with K_{f}^{m} by Δ_{0}. Since the natural projection of K_{f}^{m} onto K_{∞}^{m} has compact kernel, Δ_{0} is discrete in K_{f}^{m}, it follows that the projection of Δ_{0} into K_{∞}^{m} is a discrete subgroup in a real vector space and, therefore, it is a finitely generated abelian group. Also, because K_{f}^{m} is torsion-free, Δ_{0} is isomorphic to its projection. If x is any element in Δ, then there exists an S-adic unit ξ such that ξx ∈ Δ_{0}. Thus Δ is a finitely generated O_{S}-module. Also Δ is without torsion because it is contained in a K-vector space and O_{S} ⊂ K. Now (5.3), with I_{1}, …, I_{r} being fractional ideals, follows from Proposition 5.3. Note that one can choose n ∈ ℤ such that nI_{i} ⊂ O_{S} for all i; thus, replacing a_{i} with \frac{1}{n}a_{i} we can assume that I_{1}, …, I_{r} are integer ideals. It remains to note that the discreteness of Δ forces a_{1}^{(v)}, …, a_{r}^{(v)} to be linearly independent over K_{v}^{m} for any v ∈ S; thus one can complete {a_{1}, …, a_{r}} to a basis {a_{1}, …, a_{m}} of the free K_{S}-module K_{S}^{m}, and take g to be the element sending the standard basis of K_{S}^{m} to {a_{1}, …, a_{m}}. □

5.5. Remark. Since Cl_{S}(K) is a finite group, there exists a finite set of integer ideals {J_{1}, …, J_{k}}, where k is the number of elements in Cl_{S}(K), representing the ideal classes of Cl_{S}(K). So, if Δ is as in Proposition 5.4, there exist α_{i} ∈ K such that I_{i}α_{i} ∈ {J_{1}, …, J_{k}} for all i. Therefore, arguing as in the proof of Proposition 5.4, one can see that the ideals I_{1}, …, I_{r} in (5.3) can be chosen from the apriori fixed set {J_{1}, …, J_{k}}.

5.6. It follows from Proposition 5.4 that the set of discrete O_{S}-submodules of K_{S}^{m} can be identified with 𝔐(K_{S}, O_{S}, m) as defined in (4.2). It also follows that any Δ ∈ 𝔐(K_{S}, O_{S}, m) is a lattice in K_{S}Δ. Let us now define cov(Δ) to be the covolume of Δ in K_{S}Δ with respect to the volume on L = K_{S}Δ normalized as in §5.2.

The following lemma shows how one can explicitly compute covolumes:

Lemma. Let Δ = I_{1}a_{1} ⊕ ⋯ ⊕ I_{r}a_{r} ∈ 𝔐(K_{S}, O_{S}, m), where I_{1}, …, I_{r} ∈ P_{S}(K) and a_{1}, …, a_{r} ∈ K_{S}^{m}. Then

cov(I_{1}a_{1} ⊕ ⋯ ⊕ I_{r}a_{r}) = (\sqrt{D_{K}})^{r}N(I_{1} ⋯ I_{r})c(a_{1} ∧ ⋯ ∧ a_{r}) .

Proof. First consider the special case Δ = O_{S}a_{1} ⊕ ⋯ ⊕ O_{S}a_{r}. Put L = K_{S}Δ, define a basis (b_{1}, …, b_{r}) of L as in (5.2ab), and then complete it to a basis (b_{1}, …, b_{m}) of the whole space K_{S}^{m}. Also let h_{v} ∈ GL(m, K_{v}) be such that h_{v}b_{i}^{(v)} = e_{i}^{(v)} for all 1 ≤ i ≤ m, where e_{1}^{(v)}, …, e_{m}^{(v)} is the standard basis of K_{v}^{m}. It follows from the definition of the measure on L that the map h : L → h(L), where h = (h_{v}) ∈ GL(m, K_{S}), is measure preserving. Since the map ⋀^{r} h : ⋀^{r} K_{S}^{m} → ⋀^{r} K_{S}^{m} preserves the content c on ⋀^{r} K_{S}^{m}, we may reduce the problem to the case b_{i} = e_{i} for all 1 ≤ i ≤ m. Let φ ∈ GL(r, K_{S}) be such that φ(e_{i}) = a_{i} for all 1 ≤ i ≤ r. Then since a_{1} ∧⋯∧a_{r} = det(φ)e_{1} ∧⋯∧e_{r}, we get

cov(Δ) = cov(φ(O_{S}^{r})) = c(det φ) cov(O_{S}^{r}) = (\sqrt{D_{K}})^{r}c(a_{1} ∧ ⋯ ∧ a_{r}) .

For arbitrary I_{1}, …, I_{r}, one can choose an element α ∈ O_{S} such that αI_{i} ⊂ O_{S} for all i. Since [O_{S} : αI_{i}] = N(αI_{i}), one has

[(O_{S}a_{1} ⊕⋯⊕O_{S}a_{r}) : (I_{1}αa_{1} ⊕⋯⊕I_{r}αa_{r})] = N(α^{r}I_{1} ⋯ I_{r}) = N(αO_{S})^{r}N(I_{1} ⋯ I_{r}).

Also, in view of (5.1),

c(α) = \frac{cov(αO_{S})}{cov(O_{S})} = N(αO_{S}) ,

Therefore

cov(I_{1}a_{1} ⊕ ⋯ ⊕ I_{r}a_{r}) = \frac{1}{c(α)^{r}} cov(I_{1}αa_{1} ⊕ ⋯ ⊕ I_{r}αa_{r})

= \frac{1}{c(α)^{r}} N(αO_{S})^{r}N(I_{1} ⋯ I_{r}) cov(O_{S}a_{1} ⊕ ⋯ ⊕ O_{S}a_{r})

= (\sqrt{D_{K}})^{r}N(I_{1} ⋯ I_{r}) c(a_{1} ∧ ⋯ ∧ a_{r}) ,

as desired. □

5.7. Lemma 5.6 immediately implies

Corollary. For every Δ ∈ 𝔐(K_{S}, O_{S}, m), the function GL(m, O_{S}) → ℝ_{+}, g ↦ cov(gΔ), is continuous.

5.8. Corollary. If Δ, Δ′ ∈ 𝔐(K_{S}, O_{S}, m) are such that KΔ ∩ KΔ′ = {0}, then

cov(Δ + Δ′) ≤ cov(Δ) cov(Δ′).

Proof. In view of Proposition 5.3 there exist linearly independent over K elements a_{1}, …, a_{r}, b_{1}, …, b_{s} and fractional ideals I_{1}, … I_{r}, J_{1}, …, J_{s} such that Δ = I_{1}a_{1} ⊕ ⋯ ⊕ I_{r}a_{r} and Δ′ = J_{1}b_{1} ⊕ ⋯ ⊕ J_{r}b_{r}.

Since

K_{S}Δ + K_{S}Δ′ = K_{S}Δ ⊕ K_{S}Δ′,

in view of Lemma 5.6, it is enough to prove that

c(a_{1} ∧ ⋯ ∧ a_{r} ∧ b_{1} ∧ ⋯ ∧ b_{s}) ≤ c(a_{1} ∧ ⋯ ∧ a_{r})c(b_{1} ∧ ⋯ ∧ b_{s}),

which is easy to verify using the definition of content and the basic properties of the exterior product. □

5.9. In the remaining part of the section we investigate metric properties of discrete submodules of K_{S}^{m}. For x = (x^{(v)}) ∈ K_{S}^{m} let us define the norm of x by ∥x∥ =^{def} sup_{v} ∥(x^{(v)})∥_{v}. If m = 1 this norm will be denoted by | · | and will be called the absolute value.

Let us now state the following S-arithmetic version of the classical Minkowski’s Lemma:

Lemma. Let Δ ∈ 𝔐(K_{S}, O_{S}, m) be of rank r, and let B be a closed ball in K_{S}Δ (with respect to the norm ∥ · ∥) centered at 0 such that λ_{S}(B) ≥ 2^{dr} cov(Δ), where d is the degree of the number field K. Then Δ ∩ B ≠ {0}.

Proof. Since the volume λ_{S} on L = K_{S}Δ was defined by identifying L with K_{S}^{r} via the basis (5.2ab), without loss of generality we can assume that Δ is a lattice in K_{S}^{r}.

Then we can write B = B_{∞} ×B_{f}, where B_{∞} =^{def} B ∩K_{∞}^{r} and B_{f} =^{def} B ∩(∏_{v∈S_{f}} K_{v})^{r}. Note that

λ_{S}(\frac{1}{2}B_{∞} × B_{f}) = \frac{1}{2^{dr}} λ_{S}(B) ≥ cov(Δ).

Since \frac{1}{2}B_{∞} × B_{f} is closed, the above implies that there exist x, y ∈ \frac{1}{2}B_{∞} × B_{f}, x ≠ y, such that x − y ∈ Δ. This finishes the proof, since clearly x − y ∈ B as well. □

5.10. We will also need the following result:

Lemma. There exists a constant A > 1 depending only on K and S such that if x ∈ K_{S}^{m} and c(x) ≠ 0 then there exists an S-adic unit ξ with

A^{−1}c(x)^{1/ℓ} ≤ ∥ξx∥ ≤ Ac(x)^{1/ℓ},

where ℓ is the cardinality of S.

Proof. Let

H = {(a_{1}, …, a_{ℓ}) ∈ ℝ_{+}^{ℓ} | a_{1} ⋯ a_{ℓ} = 1}.

If S = {v_{1}, …, v_{ℓ}}, we define the map i : O_{S}^{∗} → H, ξ → (|ξ|_{v_{1}}, …, |ξ|_{v_{ℓ}}). By the proof of the Dirichlet Unit Theorem (see [L, Ch. V, §1]) i(O_{S}^{∗}) is a cocompact lattice in the multiplicative group H. Therefore there exists a constant A > 1 such that for any (a_{i}) ∈ H one can find ξ ∈ O_{S}^{∗} with

(5.5)  A^{−1} ≤ |ξ|_{v_{i}}a_{i} ≤ A

for all i.

Let x = (x^{(v_{i})}) ∈ K_{S}^{m} and c(x) ≠ 0. Note that the vector (\frac{∥x∥_{v_{i}}}{c(x)^{1/ℓ}} ) is in H, and one has

∥ξx^{(v_{i})}∥_{v_{i}} = |ξ|_{v_{i}}∥x^{(v_{i})}∥_{v_{i}},

and

c(x) = c(ξx)

for all ξ ∈ O_{S}^{∗}. This, in view of (5.5), implies the claim. □

5.11. Let us put

(5.5)  Ω_{S,m} =^{def} {gO_{S}^{m} | g ∈ GL(m, K_{S})} .

Since O_{S}^{m} is a lattice in K_{S}^{m}, every Λ ∈ Ω_{S,m} is a free O_{S}-module of rank m and a lattice in K_{S}^{m}.

Corollary. For any Λ ∈ Ω_{S,m} and any ρ > 0, the number of submodules of Λ with covolume ≤ ρ is finite.

Proof. It follows from Lemma 5.6 and Remark 5.5 that it is enough to prove that the number of free O_{S}-submodules of Λ with covolume < ρ is finite. If Δ is a free O_{S}-submodule with rk(Δ) = r, then ⋀^{r} Δ is a free O_{S}-submodule of rank 1 of the lattice ⋀^{r} Λ in ⋀^{r} K_{S}^{m}, and, in view of Lemma 5.6,

cov(Δ) = (\sqrt{D_{K}})^{r−1} cov(⋀^{r} Δ).

Therefore it is enough to prove that the number of rank-one free O_{S}-submodules of Λ is finite. If Δ = O_{S}a is such a submodule, then in view of Lemma 5.10 the generator a can be chosen in such a way that

∥a∥ ≤ A (\frac{ρ}{\sqrt{D_{K}}} )^{1/ℓ} .

Since Λ is discrete in K_{S}^{m}, the set of all a ∈ Λ satisfying the above inequality is finite, which proves the corollary. □

5.12. Because of lack of an appropriate reference we will prove the S-adic version of Mahler compactness criterion. Consider the group

(5.6)  GL^{1}(m, K_{S}) =^{def} {g ∈ GL(m, K_{S}) | c( det(g)) = 1} ,

consisting of λ_{S}-preserving linear automorphisms of K_{S}^{m}. Let

Ω_{S,m}^{1} =^{def} \{Λ ∈ Ω_{S,m} | cov(Λ) = (\sqrt{D_{K}})^{m}\} .

Note that O_{S}^{m} is an element of Ω_{S,m}^{1}, and its stabilizer in GL(m, K_{S}) coincides with GL(m, O_{S}) which is understood to be diagonally imbedded in GL(m, K_{S}). Note also that it follows from Artin’s Product Formula [C, Theorem 10.2.1] that GL(m, O_{S}) is contained in GL^{1}(m, K_{S}). Thus Ω_{S,m}^{1} is naturally identified with the homogeneous space GL^{1}(m, K_{S})/ GL(m, O_{S}). Since SL(m, O_{S}) is a lattice in SL(m, K_{S}) and O_{S}^{∗} is a lattice in GL(1, K_{S}), GL(m, O_{S}) is a lattice in GL^{1}(m, K_{S}).

Let us say that a set Q of lattices in R^{m}, where R is a topological ring, is separated from 0 if there exists a nonempty neighborhood B of 0 in R^{m} such that Λ ∩ B = {0} for all Λ ∈ Q.

Theorem (Mahler compactness criterion). A subset Q ⊂ Ω_{S,m}^{1} is bounded if and only if it is separated from 0.

Proof. The implication (⟹) is trivial. In order to prove the converse, note that

GL^{1}(m, K_{S}) = K_{S}^{1} ⋉ SL(m, K_{S}) and GL(m, O_{S}) = O_{S}^{∗} ⋉ SL(m, O_{S}) ,

where K_{S}^{1} = {x ∈ K_{S} | c(x) = 1}. Since O_{S}^{∗} is a cocompact lattice in K_{S}^{1}, it is enough to prove the theorem with Ω_{S,m}^{1} replaced by SL(m, K_{S})/ SL(m, O_{S}), i.e. with the set

{gO_{S}^{m} | g ∈ SL(m, O_{S})} .

It follows from the strong approximation theorem for classical groups [Kn] that

SL(m, K_{S}) = SL(m, K_{f}) SL(m, O_{S}) .

Thus every g ∈ SL(m, K_{S}) can be represented as g = g_{f}g_{l}, where g_{f} ∈ SL(m, K_{f}) and g_{l} ∈ SL(m, O_{S}). One has

(5.7)  gO_{S}^{m} ∩ K_{f}^{m} = g_{f} (O_{S}^{m} ∩ K_{f}^{m}) = g_{f}O^{m} .

Let Q̃ ⊂ SL(m, K_{S}) be such that the set of lattices {gO_{S}^{m} | g ∈ Q̃} is separated from 0. It follows from (5.7) that

(5.8)  {g_{f}O^{m} | g ∈ Q̃} is separated from 0 in K_{f}^{m} .

Note that SL(m, K_{f}) = SL(m, K_{∞})×SL(m, O_{f}), and, therefore, every g_{f} ∈ SL(m, K_{f}) can be written as g_{f} = g_{∞}g_{c}, where g_{∞} ∈ SL(m, K_{∞}) and g_{c} belongs to the compact group SL(m, O_{f}). It follows from (5.8) that {g_{∞}O^{m} | g ∈ Q̃} is separated from 0. This reduces the proof to the case when S = ∞. In this case, using Weil’s restriction of scalars functor R_{K/ℚ} [Zi, §6.1], one can identify SL(m, K_{∞}) with the group of ℝ-points G of a ℚ-algebraic subgroup of SL(N, ℂ) and SL(m, O) with G ∩ SL(N, ℤ). Now the theorem follows from Mahler’s compactness criterion for SL(N, ℝ)/ SL(N, ℤ) (see [R, Corollary 10.9]). □

6. S-arithmetic quantitative nondivergence

6.1. In this section we apply Theorem 4.4 to the triple (D, K, R) of a particular type. Namely, as in §5, we let K be a number field, choose a finite set S of valuations | · |_{v} of K containing all the Archimedean ones, and for the rest of this section take D = O_{S} and R = K_{S}. Then for any m ∈ ℕ, the set 𝔐(K_{S}, O_{S}, m), as defined in (4.2), is equal to the set of all submodules of all lattices Λ ∈ Ω_{S,m}, where Ω_{S,m} is as defined in (5.5). Let us now state the following

Lemma. The function ν : 𝔐(K_{S}, O_{S}, m) → ℝ_{+} given by ν(Δ) = cov(Δ), with cov(·) as defined in §5.6, is norm-like, with C_{ν} = 1.

Proof. Property (N1) is straightforward since Δ′ ⊂ Δ and K_{S}Δ′ = K_{S}Δ implies that Δ is a subgroup of Δ′ of finite idex and cov(Δ′) = [Δ : Δ′] cov(Δ). Property (N2) with C_{ν} = 1 follows from Corollary 5.8 with Δ′ = O_{S}γ. Finally, (N3) has already been mentioned as Corollary 5.7. □

6.2. Now define a function δ : Ω_{S,m} → ℝ_{+} by

δ(Λ) =^{def} min {c(x) | x ∈ Λ ∖ {0}} .

Note that the minimum is well defined due to Lemma 5.10 and every Λ ∈ Ω_{S,m} being discrete in K_{S}^{m}. We will use the following

Lemma. There exists a constant A > 0 depending only on K, S and m such that the following holds: for ρ > 0 and Λ ∈ Ω_{S,m} suppose there exists a submodule Δ of Λ with cov(Δ) ≤ ρ; then δ(Λ) ≤ Aρ^{1/m}.

Proof. Take ε > 0 and let B be a ball in K_{S}Δ centered at 0 of radius ε (with respect to the norm ∥ · ∥ introduced in §5.9). Then one has λ_{S}(B) ≤ const ·ε^{rℓ}, where const depends only on K, S and m, r = rk(Δ), and ℓ is the cardinality of S. By Lemma 5.9, Δ has a nontrivial intersection with B whenever const ·ε^{rℓ} ≥ 2^{dr}ρ. This shows how one can choose A such that Δ (and hence Λ) is guaranteed to contain a nonzero vector x with ∥x∥ ≤ A^{1/ℓ}ρ^{1/rℓ}, which clearly implies C(x)Aρ^{1/r} ≤ Aρ^{1/m}. □

6.3. As in §4, let us use the notation 𝔓(O_{S}, m) for the set of all nonzero primitive submodules of O_{S}^{m}.

Theorem. Let X be a Besicovitch metric space, μ a uniformly Federer measure on X, and let K and S be as above. For m ∈ ℕ, let a ball B = B(x_{0}, r_{0}) ⊂ X and a continuous map h : B̃ → GL(m, K_{S}) be given, where B̃ stands for B(x_{0}, 3^{m}r_{0}). Now suppose that for some C, α > 0 and 0 < ρ < 1 one has

(i) for every Δ ∈ 𝔓(O_{S}, m), the function cov (h(·)Δ) is (C, α)-good on B̃ with respect to μ;

(ii) for every Δ ∈ 𝔓(O_{S}, m), ∥ cov (h(·)Δ)∥_{μ,B} ≥ ρ.

Then for any positive ε ≤ ρ /\sqrt{D_{K}} one has

(6.1)  μ (\{x ∈ B | δ(h(x)O_{S}^{m}) < ε\}) ≤ mC(N_{X}D_{μ}^{2})^{m} (\frac{ε\sqrt{D_{K}}}{ρ} )^{α} μ(B) .

Note that [KLW, Theorem 4.3] is a special case (corresponding to K = ℚ and S = {∞}) of the above theorem.

Proof. To apply Theorem 4.4, one uses Lemma 6.1 which guarantees the norm-like property of cov(·), and Corollary 5.11 which implies condition (iii) of Theorem 4.4. To derive (6.1) from (4.3) it remains to observe that δ(h(x)O_{S}^{m}) < ε implies the existence of a vector x ∈ h(x)O_{S}^{m} ∖ {0} with

cov(O_{S}x) = \sqrt{D_{K}} · c(x) ≤ \sqrt{D_{K}} · ε . □

6.4. In order to interprete the above result, let us assume, as it will be the case in many applications, that the function h takes values in the group GL^{1}(m, K_{S}). Then for any x ∈ B, h(x)O_{S}^{m} belongs to Ω_{S,m}^{1}. It follows from Lemma 5.10 and Theorem 5.12 that for any positive ε, the set

Q_{ε} =^{def} {Λ ∈ Ω_{S,m}^{1} | δ(Λ) ≥ ε}

is compact, and clearly Ω_{S,m}^{1} is the union of Q_{ε} over all ε > 0. This way, Theorem 6.3 estimates, in terms of ε, the relative measure of points x ∈ B which are mapped, by x ↦ h(x)O_{S}^{m}, to the complement of Q_{ε} in Ω_{S,m}^{1}.

As an application, let us take X, μ and h of a special form. Namely, for every v ∈ S choose d_{v} ∈ ℕ, and consider X = ∏_{v∈S} K_{v}^{d_{v}}, μ = λ as defined in (0.2), and a map h = (h_{v})_{v∈S} : X → GL^{1}(m, K_{S}), where each h_{v} is a map from K_{v}^{d_{v}} to GL(m, K_{v}). We say that h is polynomial (or regular) if for every v all matrix coefficients of h_{v} and its inverse are polynomials (equivalently, if every h_{v} is the restriction of a regular map of algebraic varieties K̄_{v}^{d_{v}} → GL(m, K̄_{v}), where K̄_{v} is the algebraic closure of K_{v} ).

Theorem. Let X and h be defined as above. Then there exists α > 0 (depending only on m, d_{v} and the degrees of the maps) such that for every compact set L ⊂ Ω_{S,m}^{1} one can find positive C_{0} and τ (depending only on m, d_{v}, the degrees of the maps h_{v}, and L) such that Q_{τ} ⊃ L, and the following property holds: for any positive ε and any ball B ⊂ X one has

(6.2)  λ ({x ∈ B | h(x)O_{S}^{m} ∉ Q_{ε}}) ≤ C_{0}ε^{α}λ(B)

whenever h(B)O_{S}^{m} ∩ Q_{τ} ≠ ∅. Furthermore, if h(X)O_{S}^{m} ∩ Q_{τ} = ∅, then there exists a proper Δ ∈ 𝔓(O_{S}, m) such that h(x)(K_{S}Δ) = h(0)(K_{S}Δ) for all x ∈ X.

Proof. It was mentioned in Example 1.6 that X is Besicovitch, and in Example 1.7 that λ is uniformly Federer. Using the exterior power representation of Lemma 5.6, one can easily show that for every Δ ∈ 𝔓(O_{S}, m) the function cov (h(·)Δ) has the form ∏_{v∈S} ∥f_{v}∥_{v}, where each f_{v} is a polynomial map from K_{v}^{d_{v}} to another vector space over K_{v}. Every such function is (C′, α′)-good with uniform C′ and α′ due to Lemma 2.4, Lemma 2.1(cd) and Corollary 2.3. Thus condition (i) of Theorem 6.3 is satisfied with some C, α > 0.

Now let us take C̃ =^{def} max (mC(N_{X}D_{λ}^{2})^{m}, 1). It follows from Lemma 6.2 and Theorem 5.12 that there exists τ = τ(L) > 0 such that for any submodule Δ of any Λ ∈ L one has cov (Δ) ≥ τ\sqrt{D_{K}}. Without loss of generality we can assume that τ < \frac{1}{(2C̃)^{1/α}\sqrt{D_{K}}}. Note that L is contained in Q_{τ}, since by definition of τ one has

c(x) = \frac{1}{\sqrt{D_{K}}} cov(O_{S}x) ≥ τ for any nonzero element x of any Λ ∈ L.

If B is such that h(B)O_{S}^{m} ∩ Q_{τ} ≠ ∅, it follows from Lemma 6.2 that condition (ii) of Theorem 6.3 is satisfied with (τ/A)^{m} in place of ρ (where A is as in Lemma 6.2). Therefore one has

λ (\{x ∈ B | h(x)O_{S}^{m} ∉ Q_{ε}\}) ≤ C̃ (\frac{ε\sqrt{D_{K}}A^{m}}{τ^{m}} )^{α} λ(B)

for all ε ≤ τ^{m} /\sqrt{D_{K}}A^{m}. Replacing, if necessary, C̃ by a larger number C, we conclude that (6.2) is valid for all positive ε.

Now assume that h(x)O_{S}^{m} ∉ Q_{τ} for all x ∈ X. Take ρ =^{def} (2C̃)^{1/α}\sqrt{D_{K}} · τ < 1, write X = ⋃_{i=1}^{∞} B_{i} where B_{i} are balls centered at 0 with B_{i} ⊂ B_{i+1} for all i, and consider

𝔓_{i} =^{def} {Δ ∈ 𝔓(O_{S}, m) | ∥ cov (h(·)Δ)∥_{B_{i}} < ρ} .

Then clearly 𝔓_{i+1} ⊂ 𝔓_{i} for all i, and all these sets are finite due to Corollary 5.11. We claim that ⋂_{i=1}^{∞} 𝔓_{i} must be nonempty. Indeed, otherwise one obtains a nonempty ball B such that ∥ cov (h(·)Δ)∥_{B} ≥ ρ for every Δ ∈ 𝔓(O_{S}, m). Thus Theorem 6.3 can be applied, and one can conclude that

λ(B) = λ (\{x ∈ B | h(x)O_{S}^{m} ∉ Q_{τ} \}) ≤ C̃ (\frac{τ\sqrt{D_{K}}}{ρ} )^{α} λ(B) = \frac{1}{2}λ(B) ,

a contradiction.

Consequently, there exists a proper Δ ∈ 𝔓(O_{S}, m) such that cov (h(x)Δ) < ρ for all x ∈ X. It follows that each of the polynomials f_{v} in the aforementioned representation for cov (h(·)Δ) is bounded. Therefore f_{v} ≡ const for each v, which implies that h(x)(K_{S}Δ) does not depend on x. □

We note that to derive Theorem 0.2 from the above theorem, one needs to take L = {Λ} and h of the form x ↦ h(x)g, where g ∈ GL^{1}(m, K_{S}) is such that Λ = gO_{S}^{m}, and observe that h(B)O_{S}^{m} ∩ Q_{τ} is nonempty whenever B contains 0.

7. Invariant locally finite measures for actions of unipotent groups on homogeneous spaces

7.1. Theorem 6.4 implies results closely related to the structure of orbits and invariant measures on S-adic homogeneous spaces under the action of subgroups generated by unipotent elements (see [Rat3], [MT1–2] and [T]).

Let us recall some definitions from [MT1]. Let T be a finite set, and for every v ∈ T let K_{v} be a non-discrete locally compact field of characteristic 0. Denote by K_{T} the direct product of all K_{v}, v ∈ T . By a K_{T} -algebraic group G we mean a (formal) direct product ∏_{v∈T} G_{v} of K_{v}-algebraic groups G_{v}. The group ∏_{v∈T} G_{v}(K_{v}) will be denoted by G(K_{T}) and called the group of K_{T} -rational points of G. We will also use the simpler notations G for G(K_{T}) and G_{v} for G_{v}(K_{v}). If H is another K_{T} - algebraic group then a homomorphism φ : G → H is called K_{T} -homomorphism if φ is a product of K_{v}-homomorphisms of algebraic groups φ_{v} : G_{v} → H_{v}, v ∈ T . We preserve the same terminology for the restriction map φ : G → H. By the Zariski topology on G (respectively, G) we mean the formal product of the Zariski topologies on G_{v} (respectively, G_{v}). By a K_{T} -algebraic subgroup of G (or simply an algebraic subgroup of G) we mean a Zariski closed subgroup of G. An element g = (g_{v}) in G is unipotent if each component g_{v} ∈ G_{v} is unipotent. A subgroup of G consisting of unipotent elements is called unipotent.

Let U be a unipotent algebraic subgroup of G. Then U = ∏_{v∈T} U_{v}, where U_{v} are unipotent algebraic subgroups of G_{v}. Given v ∈ T , we denote by exp_{v} : Lie(U_{v}) → U_{v} the exponential map and by log_{v} = exp_{v}^{−1} the logarithmic map. Also we denote by Lie(U) the direct product of the Lie algebras Lie(U_{v}) of U_{v}, v ∈ T and by exp : Lie(U) → U the direct product of the maps exp_{v}. By a (rational) parametrization of U we mean a product ϕ = (ϕ_{v})_{v∈T} of surjective maps ϕ_{v} : K_{v}^{d_{v}} → U_{v}, v ∈ T , such that for every v ∈ T the map log_{v} ∘ϕ_{v} : K_{v}^{d_{v}} → Lie(U_{v}) is polynomial and ϕ_{v}(0) = e. If d_{v} is the degree of log ∘ϕ_{v} then d = max{d_{v}|v ∈ T } is called the degree of the parametrization ϕ. Clearly exp is a parametrization of U which we call trivial. We will denote by λ the Haar measure on X = ∏_{v∈T} K_{v}^{d_{v}}.

Up to the end of this section we will denote by Γ a lattice in G. Any subgroup of G acts on the homogeneous space G/Γ by left translations.

The following theorem generalizes earlier results for one-parameter real groups proved in [D1, D2] and for one-parametric ultrametric groups announced, with indications of the proof, in [MT1, Theorem 11.4] and [Rat, Theorem 9.1].

Theorem. Let G and Γ be as above, let d be a natural number, and let L be a compact subset of G/Γ. Then L is contained in a compact L_{0} with the following property: given β > 0 there exists a compact M_{β} ⊂ G/Γ such that for any y ∈ G/Γ and any parametrization ϕ : X → U of degree ≤ d of a unipotent algebraic subgroup U of G one of the following holds:

(i) If Uy ∩ L_{0} ≠ ∅ and B is a ball in X with ϕ(B)y ∩ L_{0} ≠ ∅ then

(7.1)  \frac{λ (\{x ∈ B | ϕ(x)y ∈ M_{β}\})}{λ(B)} ≥ 1 − β .

In particular, (7.1) is satisfied if y ∈ L and B contains the origin.

(ii) If Uy ∩ L_{0} = ∅, then there exists a proper closed subgroup H of G containing U such that the orbit Hy is closed and carries a finite H-invariant Borel measure.

Proof. Applying Weil’s restriction of scalars functor, we may assume without loss of generality that each K_{v} is a completion of the field of rationals ℚ, so that we have T = S and K_{T} = ℚ_{S}. Let R(G) be the direct product of the solvable radicals R(G_{v}) of G_{v}, v ∈ S. It is easy to see using Auslander’s theorem ([Rag, Theorem 8.24], [MT2, Lemma 1.11] and [Rag, Theorem 3.1]) that R(G) ∩ Γ is a cocompact lattice in R(G). Let N be a maximal subgroup in the class of all normal algebraic subgroups of G containing R(G) as a cocompact subgroup. Let N be the Zariski closure of N in G. By the general structure theory of algebraic groups [Bo], H = G/N is a ℚ_{S}- algebraic group, H = ∏_{v∈S} H_{v} where each H_{v} is a semisimple group and H_{v}(K_{v}) has no compact factors, and there exists a ℚ_{S}-homomorphism φ : G → H (where H = H(ℚ_{S})) such that φ(G) has finite index in H. Denote φ(Γ) by Σ. Since N ∩ Γ is cocompact in N, Σ is a lattice in H and the natural map φ̃ : G/Γ → H/Σ is proper. Also note that φ ∘ ϕ is a rational parametrization for φ(U) of degree depending only on the degree of ϕ and the homomorphism φ. Therefore without loss of generality we can reduce the proof to the case when every G_{v} is a semisimple group without compact factors. Furthermore, we may assume that Γ is an irreducible lattice in G. Let rank_{S} G = ∑_{p∈S} rank_{ℚ_{v}} G_{v} be the S-rank of G. If rank_{S} G = 1, then either G is a real rank-one semisimple group and the theorem is proved in [D2] (see also [D3, Remark 3.7]), or G is a p-adic Lie group and, therefore, Γ is a cocompact lattice [Ta] in G and there is nothing to prove. It remains to consider the case rank_{S} G > 1. In view of Margulis’ Arithmeticity Theorem [Zi, Theorem 10.1.12] we may assume that G is a ℚ-algebraic subgroup of SL(m), Γ = G ∩ SL(m, O_{S}) and, after eventually replacing G by its image under a ℚ-irreducible representation, we may also assume that G(ℚ) acts irreducibly on ℚ^{m}.

Writing y in the form gΓ for some g ∈ G and applying Theorem 6.4 with the map h(x) = ϕ(x)g, we get a compact L_{0} = Q_{τ} ⊃ L and constants C, α > 0 such that for any τ > 0 and any ball B ⊂ X one has

λ ({x ∈ B | ϕ(x)gO_{S}^{m} ∈ Q_{ε}}) ≥ (1 − Cε^{α})λ(B),

whenever ϕ(B)y∩L_{0} ≠ ∅. Choosing ε such that β ≥ Cε^{α}, we get a compact M_{β} =^{def} Q_{ε} satisfying (7.1).

If Uy ∩ L_{0} = ∅, then in view of Theorem 6.4 there exists a proper non-zero vector subspace V ⊂ ℚ^{m} such that K_{S}V is invariant under the action of g^{−1}Ug. (We consider ℚ^{m} diagonally imbedded in ℚ_{S}^{m}, which justifies the expression K_{S}V.) Let P be the Zariski closure of the stabilizer of V in G(ℚ) under the natural action of G(ℚ) on ℚ^{m}. Since G(ℚ) acts irreducibly on ℚ^{m}, P is a proper ℚ-algebraic subgroup of G. Therefore so is the subgroup P_{u} of P spanned by all unipotent elements of P. It is easy to see that g^{−1}Ug ⊂ P_{u}(ℚ_{S}). By Borel–Harish-Chandra theorem P_{u}(O_{S}) is a lattice in P_{u}(ℚ_{S}). Therefore the group H = gP_{u}(ℚ_{S})g^{−1} satisfies the requirements of the formulation of the theorem. □

We remark that to derive Theorem 0.3 from the above theorem it suffices to take L = {y} and, in the case Uy∩L_{0} ≠ ∅, choose R such that the intersection of ϕ(B(0, R)) with L_{0} is nonempty,

7.2. Note that in the above theorem the measure λ can be replaced by a Haar measure σ on U after appropriately choosing a metric on U. Indeed, given a bijective parametrization ϕ : X → U, let ϕ_{∗}λ be the pushforward of λ to U via ϕ. It is enough to find a ϕ such that σ = ϕ_{∗}λ. If U is abelian, ϕ can be taken to be the trivial parametrization (because exp is an isomorphism of locally compact groups). In the general case, since U is unipotent and algebraic, there exist abelian algebraic subgroups U_{1}, …, U_{s} of U such that the map ψ : U_{1} × ⋯ × U_{s} → U, ψ(u_{1}, …, u_{s}) = u_{1} ⋯ u_{s}, is bijective, and for any i, 1 ≤ i < s, one knows that U_{1} ⋯ U_{i} is a normal subgroup of U and U_{1} ⋯ U_{i+1}/U_{1} ⋯ U_{i} is a central subgroup of U/U_{1} ⋯ U_{i}. For any j let ϕ_{j} be the trivial parametrization of the abelian group U_{j}. Then a simple computation shows that ϕ = ψ ∘ (ϕ_{1}, …, ϕ_{s}) is such that σ = ϕ_{∗}λ, proving the claim. Using the bijective map ϕ we translate the metric from X to U. In view of the preceding discussions, Theorem 7.1 immediately implies the following

Theorem. Let G and Γ be as in Theorem 7.1. Then every compact L ⊂ G/Γ is contained in a compact L_{0} with the following property: given β > 0 there exists a compact subset M_{β} of G/Γ such that for any y ∈ G/Γ and any unipotent algebraic subgroup U of G one of the following is satisfied:

(i) If Uy ∩ L_{0} ≠ ∅ and B is a ball in U such that ϕ(B)y ∩ L_{0} ≠ ∅, then

(7.2)  \frac{σ (\{u ∈ B | uy ∈ M_{β}\})}{σ(B)} ≥ 1 − β ,

where σ is the Haar measure on U. In particular, (7.2) holds if y ∈ L and e ∈ B.

(ii) If Uy ∩ L_{0} = ∅, then there exists a closed proper subgroup H of G containing U such that the orbit Hy is closed and carries a finite H-invariant Borel measure.

7.3. We can now proceed with the

Proof of Theorem 0.4. In order to prove the theorem, as in the real case considered in [D1], it is enough to find a function h ∈ L^{1}(G/Γ, μ) which is G-invariant and h(y) > 0 for μ-a.e. y ∈ G/Γ. Indeed, if h is such a function, then the sets

Y_{i} = \{y ∈ G/Γ | h(y) ≥ \frac{1}{i}\}

satisfy the conditions in the formulation of the theorem.

Among the unipotent algebraic subgroups in H we fix a maximal one and denote it by U. It is well known that the minimal normal subgroup of H containing U coincides with H itself (see [Bo]). In view of the Mautner phenomenon for products of real and p-adic Lie groups [MT2, Proposition 2.1], if f ∈ L^{1}(G/Γ, μ) is U-invariant, then there exists an G-invariant μ-integrable function on G/Γ which coincides with f almost everywhere. Therefore it is enough to prove the theorem for H = U.

Let μ = ∫_{(A,ρ)} μ_{a} dρ be the decomposition of μ into U-invariant ergodic locally finite measures, where (A, ρ) is a measure space parametrizing the ergodic components μ_{a}. For almost every y ∈ G/Γ there exists a well defined ergodic component μ_{a(y)}, where a(y) ∈ A, whose support contains y. Further on, when it does not lead to confusion, we will identify U with its Lie algebra via exp. Let f be a positive continuous μ-integrable function on G/Γ. In view of [MT1, Proposition 7.1, Lemma 7.2 and formula (4) in §5] and [Tem, Ch. 6], the following version of Birkhoff individual ergodic theorem holds: there exists a sequence of linear parametrizations ϕ_{i} : U → U such that for any relatively compact measurable neighborhood B of e in U we have U = ⋃_{i} B_{i}, where B_{i} = ϕ_{i}(B), and for almost all y ∈ G/Γ

lim_{n→∞} \frac{1}{σ(B_{n})} ∫_{B_{n}} f(gy) dσ(g) = ∫_{G/Γ} f(z) dμ_{a(y)}(z) ,

where σ is the Haar measure on U. Furthermore, the limit function

f^{∗}(y) =^{def} ∫_{G/Γ} f(z) dμ_{a(y)}(z)

is μ-integrable and U-invariant. Note that

\frac{1}{σ(B_{n})} ∫_{B_{n}} f(gy) dσ(g) = \frac{1}{σ(B)} ∫_{B} f(ϕ_{n}(g)y) dσ(g) .

It follows from Theorem 7.1 and the fact that σ = ϕ_{∗}λ for some rational parametrization ϕ of U that there exists a compact M ⊂ G/Γ such that

\frac{σ(\{g ∈ B | ϕ_{i}(g)y ∈ M\})}{σ(B)} > \frac{1}{2}

for all i. Since f is positive, the above formula implies that f^{∗}(y) > 0 μ-a.e., which, in view of the preceding discussion, completes the proof of the theorem. □

7.4. Remark. Using the methods from [MT2] it is easy to see that Theorems 7.1, 7.2 and 0.4 remain valid for the larger class of so-called almost linear groups (that is, when G is a finite direct product of a connected real Lie group and finite central extensions of closed linear p-adic groups) as well as for the class of groups treated in [Rat3].

8. S-arithmetic Diophantine approximation

8.1. In this section we state the main Diophantine result of the present paper, of which Theorems 0.5, 0.6 and 0.7 are special cases. Here and for the rest of the paper we fix a set S of cardinality ℓ consisting of distinct normalized valuations of ℚ, denote by ℚ_{v} the completion of ℚ with respect to v ∈ S (so that ℚ_{v} = ℚ_{p} if v is the p-adic valuation, and ℚ_{v} = ℝ if v is the Archimedean one), and let ℚ_{S} =^{def} ∏_{v∈S} ℚ_{v}. We remark that similarly, with some modifications, it is possible to treat the case of an arbitrary finite extension K of ℚ; however we chose to only discuss the case K = ℚ for the sake of making our terminology and results more coincise and transparent.

We will interprete elements

y = (y^{(v)})_{v∈S} = (y_{1}, …, y_{n})

of ℚ_{S}^{n}, where y^{(v)} = (y_{1}^{(v)}, …, y_{n}^{(v)}) ∈ ℚ_{v}^{n} and y_{i} = (y_{i}^{(v)}) ∈ ℚ_{S}, as linear forms on ℚ_{S}^{n}, and will study their values y · q = y_{1}q_{1} + ⋯ + y_{n}q_{n} at integer points q = (q_{1}, …, q_{n}). The approximation properties of our interest will be related to these values being close (in terms of the S-adic absolute value | · | on ℚ_{S}) to integers. Alternatively, one could consider a dual case when one approximates (in terms of the S-adic norm) y ∈ ℚ_{S}^{n} by rational vectors. See [KLW] and [KM, Remark 6.2] for a discussion of this set-up in the real case.

For y, q as above and for q_{0} ∈ ℤ, it will be convenient to use the notation

ỹ =^{def} (1, y_{1}, …, y_{n}) and q̃ =^{def} (q_{0}, q_{1}, …, q_{n}) ,

so that q_{0} + y_{1}q_{1} + ⋯ + y_{n}q_{n} = q_{0} + y · q is written as ỹ · q̃. Also, by the absolute value | · | of integers and the norm ∥ · ∥ of integer vectors we will always mean those coming from the infinite valuation. Hopefully it will cause no confusion.

8.2. A natural starting point in the theory of simultaneous Diophantine approximation is usually a Dirichlet-principle-type result. Let us work it out. The goal is to find the optimal exponent β such that for any y ∈ ℚ_{S}^{n} and any N > 0 one is guaranteed to have two different integer vectors q̃_{1}, q̃_{2} of norm ≤ N such that ỹ · q̃_{1} and ỹ · q̃_{2} are at most const(y)N^{−β} apart.

It turns out that the answer depends on whether or not S contains the Archimedean valuation v = ∞. Indeed, note that one has

(8.1)  |ỹ^{(v)} · q̃|_{v} ≤ max(∥y^{(v)}∥_{v}, 1)

for any ultrametric v and any integer vectors q̃. Therefore:

• If all the valuations in S are ultrametric, the (2N + 1)^{n+1} values of ỹ · q̃ for all q̃ of norm ≤ N are in the ball of radius max(∥y∥, 1) in ℚ_{S}. The latter can be partitioned into const ·N^{n+1} balls of radius const(y)N^{− \frac{n+1}{ℓ}}. Thus for any y ∈ ℚ_{S}^{n} and any N > 0 one can find q̃ ∈ ℤ^{n+1} ∖ {0} with ∥q̃∥ ≤ N and

(8.2)  |ỹ · q̃| ≤ const(y) · N^{− \frac{n+1}{ℓ}} .

• If v = ∞, there is clearly no universal upper bound similar to (8.1). However one can for any given q ∈ ℤ^{n} choose q_{0} ∈ ℤ such that

(8.1_{∞})  |ỹ^{(∞)} · q̃|_{∞} = |q_{0} + y^{(∞)} · q|_{∞} ≤ 1 .

Thus, taking all q of norm ≤ N, one is only guaranteed to have (2N + 1)^{n} values of ỹ · q̃ in the ball of radius max(∥y∥, 1) in ℚ_{S}. Partitioning it into const ·N^{n} balls of radius const(y)N^{− \frac{n}{ℓ}}, one gets a nonzero integer vector q̃ with ∥q∥ ≤ N and

(8.2_{∞})  |ỹ · q̃| = |q_{0} + y · q| ≤ const(y)N^{− \frac{n}{ℓ}} for some q_{0} ∈ ℤ .

Note that the absolute value of q_{0} above, and hence the norm of q̃, is bounded from above by const(y)N.

It will be convenient to define

i_{S} =^{def} \{ \begin{matrix}1 if ∞ ∉ S \\ 0 if ∞ ∈ S . \end{matrix}

Then it follows that for any y ∈ ℚ_{S}^{n} the supremum of w > 0 for which there exist infinitely many q̃ ∈ ℤ^{n+1} with

|ỹ · q̃|^{ℓ} ≤ ∥q̃∥^{−w}

is not less than n + i_{S}. On the other hand, it can be easily shown using the Borel-Cantelli Lemma that the above supremum is equal to n + i_{S} for almost every y ∈ ℚ_{S}^{n} (with respect to Haar measure λ_{S} on ℚ_{S}^{n}). Similarly to the real/p-adic situation, we say that y ∈ ℚ_{S}^{n} is very well approximable, or VWA, if the above supremum is strictly bigger than n + i_{S}; in other words, if for some ε > 0 there are infinitely many solutions q̃ ∈ ℤ^{n+1} to

(8.3)  |ỹ · q̃|^{ℓ} ≤ ∥q̃∥^{−(n+i_{S})(1+ε)} .

Note that in the case when ∞ ∈ S, any solution q̃ of (8.3) automatically satisfies

(8.3′)  |q_{0}| ≤ 1 + n∥y^{(∞)}∥_{∞}∥q∥ ,

and hence (8.3) can be replaced by

(8.3_{∞})  |ỹ · q̃|^{ℓ} ≤ ∥q∥^{−n(1+ε)} ,

agreeing with the way it was written in the introduction for S = {∞}.

8.3. The next step is to define very well multiplicatively approximable, or VWMA, vectors y ∈ ℚ_{S}^{n}. To do this, first let us replace the left hand side of (8.3) by the product of norms of all the components of ỹ · q̃; that is, write

(8.4)  c(ỹ · q̃) ≤ ∥q̃∥^{−(n+i_{S})(1+ε)} .

Even though the latter inequality is weaker than (8.3), one can still show that for any positive ε it has at most finitely many solutions for almost every y. However, if one wants to weaken it even further by replacing the norm of q̃ in (8.4) with the geometric mean of its coordinates, one needs to be careful and keep in mind the dichotomy in the Dirichlet-principle argument.

Namely, if ∞ ∉ S one can indeed replace ∥q̃∥^{n+1} by Π_{+}(q̃), and thus define y ∈ ℚ_{S}^{n} to be VWMA if for some ε > 0 there are infinitely many solutions q̃ ∈ ℤ^{n+1} to

(8.5)  c(ỹ · q̃) ≤ Π_{+}(q̃)^{−(1+ε)} ,

as was done in the introduction for S = {p}.

On the other hand, if ∞ ∈ S it seems tempting to mimic the situation for S = {∞} and define y to be VWMA if for some ε > 0 there are infinitely many q such that

(8.5′)  c(ỹ · q̃) = c(q_{0} + y · q) ≤ Π_{+}(q)^{−(1+ε)}

holds for some q_{0} ∈ ℤ. However it is not hard to see that, whenever the cardinality of S is at least 2, for any ε > 0 the set of y ∈ ℚ_{S}^{n} for which (8.5′) admits infinitely many solutions has full measure. Indeed, the trouble here comes from the fact that an upper estimate for c(ỹ · q̃) does not imply a bound similar to (8.3′), that is, a bound on q_{0} in terms of q. And one can easily show that for any fixed q the set of y ∈ ℚ_{S}^{n} for which there exists q_{0} saitsfying (8.5′) has full measure.

It follows that in order to achieve a multiplicative analogue of (8.4) in the case {∞} ⊊ S, one needs to take special precautions in the case when |q_{0}| is much bigger than the norm of q. Here is a suggested way to do it. For any y ∈ ℚ_{S}^{n} with ∞ ∈ S and any q̃ ∈ ℤ^{n+1} let us define

Ψ_{y}(q̃) =^{def} \{ \begin{matrix}1 if |q_{0}| ≤ (1 + n∥y^{(∞)}∥_{∞}) · ∥q∥ \\ |q_{0}| otherwise , \end{matrix}

and use this function to make a modification to (8.5′). Namely, in the case ∞ ∈ S we will define y ∈ ℚ_{S}^{n} to be VWMA if for some ε > 0 there are infinitely many solutions q to

(8.5_{∞})  c(ỹ · q̃) ≤ Π_{+}(q)^{−(1+ε)}Ψ_{y}(q̃)^{−ε} .

Several remarks are in order. First, as was observed before, if S = {∞}, then the second case in the definition of Ψ_{y}(q̃) never occurs, thus the above definition is equivalent to the one given in the introduction. Second, it is easily seen that infinitely many solutions to (8.4) imply infinitely many solutions to (8.5_{∞}), perhaps with a slightly different value of ε. And yet, VWMA as defined above happens to be a zero measure condition. This can be shown directly using a Borel-Cantelli argument, and it will also be an implication of Theorem 8.4 below.

8.4. Generalizing definitions given in the introduction, let us say that a measure μ on ℚ_{S}^{n} is extremal (resp., strongly extremal) if μ-almost every point of ℚ_{S}^{n} is not VWA (resp., not VWMA). Here is the main theorem of the section:

Theorem. For v ∈ S, let X_{v} be a metric space with a measure μ_{v} such that X = ∏_{v∈S} X_{v} is Besicovitch and μ = ∏_{v∈S} μ_{v} is Federer, and let f = (f^{(v)})_{v∈S}, where f^{(v)} are continuous maps from X_{v} to ℚ_{v}^{n} which are μ_{v}-good and μ_{v}-nonplanar at μ_{v}-almost every point of X_{v}. Then f_{∗}μ is strongly extremal.

It is clear from Examples 1.6 and 1.7, Lemma 2.4 and Corollary 2.3 that Theorem 0.7, and hence Theorems 0.5 and 0.6, follow from the above result.

8.5. Let us also remark that Theorem 8.4 generalizes the main result of [KLW]. Indeed, in the latter paper a certain class of measures on ℝ^{n} was introduced, and it was proved that measures from that class are strongly extremal. Specifically, following [KLW] let us say that a measure μ on F^{n}, where F is a locally compact field, is

• nonplanar if μ(L) = 0 for any proper affine subspace of F^{n};

• decaying if for for μ-a.e. y ∈ F^{n} there exist a neighborhood V of y and C, α > 0 such that all affine functions are (C, α)-good on V with respect to μ;

• friendly if it is Federer, nonplanar and decaying. Comparing this with §2.5, one easily observes that μ is decaying if and only if the

identity map F^{n} → F^{n} is μ-good at μ-almost every point. It is also not hard to see that the nonplanarity of μ forces the aforementioned identity map to be μ-nonplanar at μ-almost every point (converse is true under the additional assumption that μ is decaying). It is now clear that Theorem 8.4 immediately implies

Corollary. Let μ = ∏_{v∈S} μ_{v}, where μ_{v} is a friendly measure on ℚ_{v}^{n} for every v ∈ S. Then μ is strongly extremal.

Thus [KLW, Theorem 1.1] is a special case of Theorem 8.4. (As was mentioned before, our proof is also a generalization of the argument from [KLW], which, in turn, generalizes the one from [KM].)

8.6. It is not hard to see that many examples of friendly measures on ℝ^{n} exhibited in [KLW] can be constructed on a vector space over arbitrary locally compact valued field F. For instance, fix a valuation | · | on F and a norm ∥ · ∥ on F^{n}, and say that a map h : F^{n} → F^{n} is a contracting similitude with contraction rate ρ if h is affine, 0 < ρ < 1 and

∥h(y)∥ = ρ∥y∥ ∀ y ∈ F^{n} .

It is known, see [H, §3.1], that for any finite family h_{1}, …, h_{m} of contracting similitudes there exists a unique nonempty compact set Q, called the attractor of the family, such that

Q = ⋃_{i=1}^{m} h_{i}(Q).

Say that h_{1}, …, h_{m} as above satisfy the open set condition if there exists an open subset U ⊂ F^{n} such that

h_{i}(U) ⊂ U for all i = 1, …, m ,

and

i ≠ j ⟹ h_{i}(U) ∩ h_{j}(U) = ∅ .

J. Hutchinson [H, §5.3] proved^{3} that if h_{i}, i = 1, …, m, are contracting similitudes with contraction rates ρ_{i} satisfying the open set condition, and if s > 0 is the unique solution of ∑_{i} ϱ_{i}^{s} = 1 , called the similarity dimension of the family {h_{i}}, then the s-dimensional Hausdorff measure H^{s} of Q is positive and finite. Let us also say that the family {h_{i}} is irreducible if there does not exist a finite {h_{i}}-invariant collection of proper affine subspaces of F^{n}. The proof of [KLW, Theorem 2.3] applies verbatim and yields

Proposition. For any completion ℚ_{v} of ℚ, let {h_{1}, …, h_{m}} be an irreducible family of contracting similitudes of ℚ_{v}^{n} satisfying the open set condition, s its similarity dimension, μ the restriction of H^{s} to its attractor. Then μ is friendly (and hence strongly extremal).

Measures on ℝ^{n} obtained via the above construction have been thoroughly studied; perhaps the simplest example is given by the \frac{log 2}{log 3}-dimensional Hausdorff measure on the Cantor ternary set. Similarly one can consider ultrametric analogues of the Cantor set, for example let

Q = {∑_{k=0}^{∞} a_{k}3^{k}| a_{k} = 1, 2} ⊂ ℤ_{3} .

It is a 3-adic version of the Cantor ternary set, which also has Hausdorff dimension s = \frac{log 2}{log 3}, and it follows that almost all numbers in Q (with respect to the s-dimensional Hausdorff measure) are not VWA.

8.7. We conclude this section with the following modification of Theorem 8.4:

Theorem. For every v ∈ S, let X_{v} be a metric space with a measure μ_{v} such that X = ∏_{v∈S} X_{v} is Besicovitch and μ = ∏_{v∈S} μ_{v} is uniformly Federer, and let f = (f^{(v)})_{v∈S}, where f^{(v)} are continuous maps from X_{v} to ℚ_{v}^{n} such that for μ_{v}-a.e. x_{v} ∈ X_{v} one can find a ball B_{v} = B(x_{v}, r) ⊂ X_{v} with the following properties:

(8.6)

for some C_{v}, α_{v} > 0, any linear combination of 1, f_{1}^{(v)}, …, f_{n}^{(v)}
is (C_{v}, α_{v})-good on B(x_{v}, 3^{n+1}r) with respect to μ_{v} ,

and

(8.7)

the restrictions of 1, f_{1}^{(v)}, …, f_{n}^{(v)} to B_{v} ∩ supp μ_{v}

are linearly independent over ℚ_{v}.

Then f_{∗}μ is strongly extremal.

Reduction of Theorem 8.4 to Theorem 8.7. Let X and f be as in Theorem 8.4. First note that, replacing X_{v} by appropriate neighborhoods of its μ_{v}-generic points for each v, one can without loss of generality assume that μ is uniformly Federer. Then for any v, since f^{(v)} is μ_{v}-good at μ_{v}-a.e. point, one can for μ_{v}-a.e. x_{v} ∈ X_{v} choose a neighborhood U_{v} of x_{v} and C_{v}, α_{v} > 0 such that any linear combination of 1, f_{1}^{(v)}, …, f_{n}^{(v)} is (C_{v}, α_{v})-good on U_{v} with respect to μ_{v}. Further, since f^{(v)} is μ_{v}-nonplanar μ_{v}- almost everywhere, one can (after throwing away points from a null set) take a ball B_{v} = B(x_{v}, r) such that B(x_{v}, 3^{n+1}r) ⊂ U_{v} and (8.7) holds, and the conclusion follows. □

3Hutchinson stated his results for the case F = ℝ, but the proofs apply verbatim to the case of arbitrary locally compact valued field.

In the next two sections we present the proof of Theorem 8.7, separately considering the cases of S containing or not containing the Archimedean valuation. In both cases the core of the proof is a generalization of the correspondence between real Diophantine approximation and dynamics on real homogeneous spaces.

9. Proof of Theorem 8.7 for ∞ ∉ S

9.1. In order to prove Theorem 8.7, we are going to dynamically interprete the approximation properties of S-adic vectors defined in the previous section, similarly to the approach of [KM]. In this section we suppose that all the valuations in S are ultrametric, that is, S = {p_{1}, …, p_{ℓ}} where p_{1}, …, p_{ℓ} are distinct primes. Up to the end of this section we will work with

S^{∗} = S ∪ \{∞\}, R =^{def} ℚ_{S^{∗}} = ℚ_{S} × ℝ, and D =^{def} O_{S^{∗}} = ℤ[\frac{1}{p_{1}} , …, \frac{1}{p_{ℓ}}] .

Then to any y ∈ ℚ_{S}^{n} we associate a lattice u_{y}D^{n+1} in R^{n+1}, where u_{y} ∈ GL^{1}(n+1, R) is defined by

u_{y}^{(p_{j})} = ( \begin{matrix}1 & y^{(p_{j})} \\ 0 & I_{n} \end{matrix} ) , j = 1, …, ℓ, u_{y}^{(∞)} = I_{n+1} ,

with I_{k} standing for the k × k identity matrix. Note that the p_{j}-adic components of

vectors from u_{y}D^{n+1} are of the form ( \begin{matrix}ỹ^{(p_{j})} · q̃ \\ q \end{matrix} ), where q̃ ∈ D^{n+1}.

We need to introduce some more notation. For a vector t̃ =^{def} (t_{0}, t_{1}, …, t_{n}) ∈ ℝ^{n+1} we denote (t_{1}, …, t_{n}) by t, and let

(9.1)  t̃= ∑_{i=0}^{n} t_{i} and t = ∑_{i=1}^{n} t_{i}

(this convention will be used throughout the next two sections, so that whenever t and t, or t̃ and t̃, appear in the same context, (9.1) will be assumed). Then, given t̃ as above and another vector s = (s_{1}, …, s_{ℓ}) ∈ ℤ_{+}^{ℓ}, define g_{s,t̃} ∈ GL(n + 1, R) by (9.2)

(g_{s,t̃})^{(p_{j})} = ( \begin{matrix}p_{j}^{−s_{j}} & 0 \\ 0 & I_{n} \end{matrix} ) , j = 1, …, ℓ, (g_{s,t̃})^{(∞)} = diag(e^{−t_{0}}, e^{−t_{1}}, …, e^{−t_{n}}) .

The next lemma shows how a good approximation for y in the sense of (8.4) gives rise to a translation of u_{y}D^{n+1} by g_{s,t̃} for some s, t̃, so that δ(g_{s,t̃}u_{y}D^{n+1}) is small. This allows one to use Theorem 6.3 to derive the needed measure estimate.

Lemma. Let ε > 0, y ∈ ℚ_{S}^{n} and q̃ ∈ ℤ^{n+1} be such that (8.5) holds. For i = 0, 1, …, n define t_{i} > 0 by

(9.3a)  |q_{i}|_{+} = Π_{+}(q̃)^{− \frac{ε}{n+1}} e^{t_{i}} ,

and let

(9.3b)  γ = \frac{ε}{(n + 1)(1 + ε)} .

Then there exists s = (s_{1}, …, s_{ℓ}) ∈ ℤ_{+}^{ℓ} such that

(9.3c)  δ(g_{s,t̃}u_{y}D^{n+1}) ≤ \sqrt{n + 1}e^{−γt̃}

and

(9.3d)  ∏_{j=1}^{ℓ} p_{j}^{s_{j}} ≤ et̃ < ∏_{j=1}^{ℓ} p_{j}^{s_{j}+1}

Proof. Multiplying equalities (9.3a), we get

(9.3e)  et̃ = Π_{+}(q̃)^{1+ε}

and

e^{−t_{i}}|q_{i}| ≤ e^{−t_{i}}|q_{i}|_{+} ≤_{(9.3a)} Π_{+}(q̃)^{− \frac{ε}{n+1}} =_{(9.3be)} e^{−γt̃}, i = 0, 1, …, n ,

hence ∥(g_{s,t̃})^{(∞)}u_{y}^{(∞)}q̃∥_{∞} = ∥(g_{s,t̃})^{(∞)}q̃∥_{∞} ≤ \sqrt{n + 1}e^{−γt̃}.

Now let us define s_{j}, j = 1, …, ℓ, inductively by

(9.4)  p_{j}^{s_{j}} ≤ min (\frac{et̃}{∏_{i=1}^{j−1} p_{i}^{s_{i}}} , \frac{1}{|ỹ^{(p_{j})} · q̃|_{p_{j}}} ) < p_{j}^{s_{j}+1}

(where if j = 1 we set ∏_{i=1}^{j−1} p_{i}^{s_{i}} = 1). This, in particular, implies that

|p_{j}^{−s_{j}} ỹ^{(p_{j})} · q̃|_{p_{j}} = p_{j}^{s_{j}}|ỹ^{(p_{j})} · q̃|_{p_{j}} ≤ 1

for each j. Taking into account that |q_{i}|_{p_{j}} ≤ 1 for all i and j, one concludes that c(g_{s,t̃}u_{y}q̃) ≤ \sqrt{n + 1}e^{−γt̃}.

It remains to check that inequalities (9.3d) are satisfied. Taking j = ℓ in (9.4) immediately implies the lower estimate. To prove the upper estimate, let us consider two cases:

• If for some j the minimum in (9.4) is equal to the first of the quantities compared, then clearly

et̃ < p_{j}^{s_{j}+1}∏_{i=1}^{j−1} p_{i}^{s_{i}} ≤ ∏_{j=1}^{ℓ} p_{j}^{s_{j}+1} .

• Otherwise, it follows that |ỹ^{(p_{j})} · q̃|_{p_{j}} > p_{j}^{−(s_{j}+1)} for all j, and to derive the desired estimate it remains to notice that (8.5), in view of (9.3e), can be rewritten as c(ỹ · q̃) = ∏_{j=1}^{ℓ} |ỹ^{(p_{j})} · q̃|_{p_{j}} ≤ e^{−t̃}. □

9.2. Corollary. Assume that y ∈ ℚ_{S}^{n} is VWMA. Then for some c, γ > 0 there are infinitely many t̃ ∈ ℤn+1_{+} and s ∈ ℤℓ_{+} such that

(9.5a)  e^{−(n+1)} ∏_{j=1}^{ℓ} p_{j}^{s_{j}} ≤ et̃ < ∏_{j=1}^{ℓ} p_{j}^{s_{j}+1}

and

(9.5b)  δ(g_{s,t̃}u_{y}D^{n+1}) ≤ ce^{−γt̃}.

Proof. By definition, for some ε > 0 there are infinitely many solutions q̃ ∈ ℤ^{n+1} of (8.5). Therefore, by the above lemma and with γ as in (9.3b), there exists an unbounded set of t̃ ∈ ℝn+1_{+} such that (9.3c) holds for some s ∈ ℤℓ_{+} satisfying (9.3d). Denote by [t̃] the vector consisting of integer parts of t_{i}, then clearly the ratio of δ(g_{s,t̃}u_{y}D^{n+1}) and δ(g_{s,[t̃]}u_{y}D^{n+1}) is bounded from above by some uniform constant. Thus, replacing t̃ by [t̃], for some c > 0 one gets infinitely many solutions t̃ ∈ ℤn+1_{+} of (9.5b), with et̃ being smaller than before by at most a factor of e^{n+1}, hence (9.5a). □

9.3. Corollary. Let X be a Besicovitch metric space and μ a uniformly Federer measure on X. Suppose we are given a continuous map f : X → ℚ_{S}^{n} such that for μ-a.e. x_{0} ∈ X there exist a ball B = B(x_{0}, r) and constants C, α, ρ with the following property: for any Δ ∈ 𝔓(D, n + 1) and any s ∈ ℤℓ_{+}, t̃ ∈ ℤn+1_{+} satisfying (9.5a), one has

(9.6a) the function x ↦ cov (g_{s,t̃}u_{f(x)}Δ) is (C, α)-good on B(x_{0}, 3^{n+1}r) w.r.t. μ ,

and

(9.6b)  sup_{x∈B ∩ supp μ} cov (g_{s,t̃}u_{f(x)}Δ) ≥ ρ .

Then f_{∗}μ is strongly extremal.

Proof. Applying Theorem 6.3, with h(x) = g_{s,t̃}u_{f(x)} and m = n + 1, we conclude that

μ (\{x ∈ B | δ(g_{s,t̃}u_{f(x)}D^{n+1}) < ce^{−γt̃}\}) ≤ (n + 1)C(N_{X}D_{μ}^{2})^{n+1} (\frac{ce^{−γt̃}}{ρ})^{α} μ(B)

whenever ce^{−γt̃} ≤ ρ and (9.5a) holds. Note that for fixed t̃, the number of different s ∈ ℤℓ_{+} satisfying (9.5a) is at most const ·t̃^{ℓ−1}. Therefore the sum (over all integer s, t̃ for which (9.5a) holds) of measures of sets {x ∈ B | δ(g_{s,t̃}u_{f(x)}D^{n+1}) < ce^{−γt̃}} is finite for every c, γ > 0. An application of the Borel-Cantelli Lemma shows that for every c, γ > 0 and μ-a.e. x ∈ B, and hence for μ-a.e. x ∈ B and all c, γ > 0, there are at most finitely many integer solutions s, t̃ to (9.5ab). Corollary 9.2 then implies that f(x) is not VWMA for μ-a.e. x ∈ B. □

9.4. We are now ready for the

Proof of Theorem 8.7, the case ∞ ∉ S. Recall that we are given the balls B_{v} ⊂ X_{v}, v ∈ S, which will be referred to as B_{1}, …, B_{ℓ}, and measures μ_{v} on X_{v}, which we will call μ_{1}, …, μ_{ℓ}. We will take B to be equal to ∏_{j=1}^{ℓ} B_{j} (recall that we are using the product metric on X) and show that it satisfies the assumptions of Corollary 9.3. Thus we need to have explicit expressions for functions x ↦ cov (g_{s,t̃}u_{f(x)}Δ).

Recall that any nonzero submodule Δ ⊂ D^{n+1} of rank r can be written in the form (5.3), and, in view of Lemma 5.6, D_{ℚ} = 1 and D being a principal ideal ring, we know that cov(Δ) = c(w), where w =^{def} a_{1} ∧ ⋯ ∧ a_{r} and a_{1}, …, a_{r} are nonzero elements of D. It will be convenient to use the standard basis e_{0}, e_{1}, …, e_{n} of R^{n+1}, where

e_{i} = (e_{i}^{(v)})_{v∈S^{∗}} = (e_{i}^{(p_{1})}, …, e_{i}^{(p_{ℓ})}, e_{i}^{(∞)})

for each i = 0, 1, …, n. Similarly, we will use the standard basis {e_{I} | I ⊂ {0, 1, …, n}}

of ⋀ R^{n+1}, where we let e_{I} =^{def} e_{i_{1}} ∧ ⋯ ∧ e_{i_{r}} ∈ ⋀^{r}(R^{n+1}) for I = {i_{1}, …, i_{r}} ⊂

{0, …, n}, i_{1} < i_{2} < ⋯ < i_{r}. Thus we can write w as above in the form w = ∑_{I⊂{0,...,n}} w_{I}e_{I}, where w_{I} ∈ D.

Now let us see how the coordinates of w as above change under the action of g_{s,t̃}u_{f(x)}. Note that:

• u_{f(x)}^{(∞)} is trivial, and each e(∞)_{I} is an eigenvector of (g_{s,t̃})^{(∞)} with eigenvalue

e^{−t_{I}}, where t_{I} =^{def} ∑_{i∈I} t_{i};

• the action of u_{f(x)}^{(p_{j})} leaves e_{0}^{(p_{j})} invariant and sends e_{i}^{(p_{j})} to e_{i}^{(p_{j})} + f_{i}^{(p_{j})}(x)e_{0}^{(p_{j})},

i = 1, …, n, and each e(p_{j})_{I} is an eigenvector of (g_{s,t̃})^{(p_{j})} with eigenvalue 1 if 0 ∉ I and p^{−s_{j}} otherwise; in other words,

(9.7) (g_{s,t̃}u_{f(x)}e_{I})^{(p_{j})} = \{ \begin{matrix}p_{j}^{−s_{j}}e(p_{j})_{I} if 0 ∈ I \\ e(p_{j})_{I} + p_{j}^{−s_{j}} ∑_{i∈I} ±f_{i}^{(p_{j})}(x)e(p_{j})_{I∪\{0\}∖\{i\}} otherwise . \end{matrix}

Therefore one has (g_{s,t̃}u_{f(x)}w)^{(∞)} = ∑_{I} e^{−t_{I}} w_{I}e(∞)_{I} and

(9.8) (g_{s,t̃}u_{f(x)}w)^{(p_{j})} = ∑_{0∉I} w_{I}e(p_{j})_{I} + p_{j}^{−s_{j}} ∑_{0∈I} (w_{I} + ∑_{i∉I} ±w_{I∪{i}∖{0}}f_{i}^{(p_{j})}(x))e(p_{j})_{I}

for j = 1, …, ℓ.

In particular, real components of all the coordinates of g_{s,t̃}u_{f(x)}w are constant, and

p_{j}-adic components are linear combinations of 1, f_{1}^{(p_{j})}, …, f_{n}^{(p_{j})}. Condition (9.6a) then immediately follows from Lemma 2.1(bc), (8.6) and Corollary 2.3. On the other hand, for any j = 1, …, ℓ one can use (8.7) and the compactness of the unit sphere in ℚ_{p_{j}}^{n+1} to find ρ_{j} > 0 such that for any a = (a_{0}, a_{1}, …, a_{n}) ∈ ℚ_{p_{j}}^{n+1} one has

(9.9)  sup_{x∈B_{j} ∩ supp μ_{j}} |a_{0} + a_{1}f_{1}^{(p_{j})}(x) + ⋯ + a_{n}f_{n}^{(p_{j})}(x)|_{p_{j}} ≥ ρ_{j}∥a∥_{p_{j}} .

It remains to notice that all the components of w necessarily appear in the second sum in (9.8) (that is, the sum of terms with 0 ∈ I). Therefore (9.8) and (9.9) imply

(9.10)  sup_{x∈B_{j} ∩ supp μ_{j}} ∥(g_{s,t̃}u_{f(x)}w)^{(p_{j})}∥_{p_{j}} ≥ ρ_{j}p_{j}^{s_{j}} max_{I} |w_{I}|_{p_{j}} ,

and hence

sup_{x∈B ∩ supp μ} c(g_{s,t̃}u_{f(x)}w) ≥ (∏_{j=1}^{ℓ} ρ_{j}p_{j}^{s_{j}} max_{I} |w_{I}|_{p_{j}} ) max_{I} e^{−t_{I}} |w_{I}|_{∞}

≥ (∏_{j=1}^{ℓ} ρ_{j}) e^{−t̃} (∏_{j=1}^{ℓ} p_{j}^{s_{j}} ) max_{I} c(w_{I}) ≥_{(9.5a)} ∏_{j=1}^{ℓ} \frac{ρ_{j}}{p_{j}} .

Condition (9.6b) is thus established, and the theorem follows. □

10. Proof of Theorem 8.7 for ∞ ∈ S

10.1. In this section we suppose that S = {p_{1}, …, p_{ℓ−1}, ∞}, where p_{1}, …, p_{ℓ−1} are distinct primes. In this case there is no need to artificially add the infinite valuation to S; that is, we will now work with

R =^{def} ℚ_{S} and D =^{def} O_{S} = ℤ[\frac{1}{p_{1}} , …, \frac{1}{p_{ℓ−1}} ] .

The element u_{y} ∈ GL^{1}(n + 1, R) can be simply defined by u_{y} =^{def} ( \begin{matrix}1 & y \\ 0 & I_{n} \end{matrix} ), so that

u_{y}D^{n+1} = \{( \begin{matrix}ỹ · q̃ \\ q \end{matrix} )| q̃ ∈ D^{n+1}\} .

The definition (9.2) of the diagonal element g_{s,t̃} ∈ GL(n + 1, R) given in the previous section will still be valid, except for s now having ℓ − 1 components.

Now let us recall that the definition of VWMA vectors is more complicated in this case than the one used in the previous section; in fact, the set of VWMA vectors can be split into two parts, according to the two alternatives in the definition of Ψ_{y}. It will be convenient to say that y ∈ ℚ_{S}^{n} is VWMA_{≤} if for some positive ε there are infinitely many solutions q̃ to

(10.1≤)  c(ỹ · q̃) ≤ Π_{+}(q)^{−(1+ε)} and |q_{0}| ≤ (1 + n∥y^{(∞)}∥_{∞}) ∥q∥ ,

and that it is VWMA_{>} if for some ε there are infinitely many solutions to

(10.1>)  c(ỹ · q̃) ≤ Π_{+}(q)^{−(1+ε)}|q_{0}|^{−ε} and |q_{0}| > (1 + n∥y^{(∞)}∥_{∞}) ∥q∥ .

Clearly y is VWMA if and only if it is either VWMA_{≤} or VWMA_{>}. Our strategy will be as follows: we will modify the dynamical approach of the previous section to treat the first case, and use the conclusion of the “∞ ∉ S” case of Theorem 8.7 to take care of the second case.

Here is a replacement for Lemma 9.1.

Lemma. Let ε > 0, y ∈ ℚ_{S}^{n} and q̃ ∈ ℤ^{n+1} be such that (10.1≤) holds. For i = 1, …, n define t_{i} > 0 by

(10.2a)  |q_{i}|_{+} = Π_{+}(q)^{− \frac{ε}{n+1}} e^{t_{i}} ,

and let

(10.2b)  γ = \frac{ε}{n + 1 + nε} .

Then there exist s = (s_{1}, …, s_{ℓ−1}) ∈ ℤ_{+}^{ℓ−1} and t_{0} ∈ ℝ such that

(10.2c)  δ(g_{s,t̃}u_{y}D^{n+1}) ≤ \sqrt{n + 1}e^{−γt} ,

(10.2d)  −t ≤ t_{0} ≤ t + ln (1 + 2n∥y^{(∞)}∥_{∞}) ,

and

(10.2e)  ∏_{j=1}^{ℓ−1} p_{j}^{s_{j}} ≤ et̃ < ∏_{j=1}^{ℓ−1} p_{j}^{s_{j}+1} .

Proof. As in the proof of Lemma 9.1, we consider the product of equalities (10.2a), namely

(10.2f)  e^{t} = Π_{+}(q)^{1+ \frac{n}{n+1} ε} ,

and then write

(10.2g)  e^{−t_{i}}|q_{i}| ≤ e^{−t_{i}}|q_{i}|_{+} ≤_{(10.2a)} Π_{+}(q)^{− \frac{ε}{n+1}} =_{(10.2bf)} e^{−γt}, i = 1, …, n .

After that define t_{0} by

(10.3)  e^{−t_{0}} =^{def} min (e^{t}, \frac{e^{−γt}}{|ỹ^{(∞)} · q̃|_{∞}} ) .

It follows that |e^{−t_{0}} ỹ^{(∞)} · q̃|_{∞} ≤ e^{−γt}, hence ∥(g_{s,t̃})^{(∞)}u_{y}^{(∞)}q̃∥_{∞} ≤ \sqrt{n + 1}e^{−γt}. The lower estimate in (10.2d) is immediate from (10.3), while the upper estimate clearly holds if the minimum in (10.3) is equal to e^{t}, and otherwise one has

e^{−t_{0}} = \frac{e^{−γt}}{|q_{0} + y^{(∞)} · q|_{∞}} ≥_{(10.1≤)} \frac{e^{−γt}}{(1 + 2n∥y^{(∞)}∥_{∞}) ∥q∥} ≥_{(10.2g)} \frac{e^{−t}}{1 + 2n∥y^{(∞)}∥_{∞}} .

Now that all the components of t̃ are chosen, we can define s_{j}, j = 1, …, ℓ − 1, as in (9.4). After that one can verify, following the lines of the proof of Lemma 9.1, that

∥(g_{s,t̃})^{(p_{j})}u_{y}^{(p_{j})}q̃∥_{p_{j}} = max ( p_{j}^{s_{j}}|ỹ^{(p_{j})} · q̃|_{p_{j}}, 1) ≤ 1

for each j, so that c(g_{s,t̃}u_{y}q̃) ≤ \sqrt{n + 1}e^{−γt}, and that inequalities (10.2e) are satisfied. □

10.2. Corollary. Assume that y ∈ ℚ_{S}^{n} is VWMA_{≤}. Then for some c_{0}, c, γ > 0 there are infinitely many t̃ ∈ ℤ × ℤn_{+} and s ∈ ℤℓ−1_{+} satisfying

(10.4a)  −t − 1 ≤ t_{0} ≤ t + c_{0} ,

(10.4b)  e^{−(n+1)} ∏_{j=1}^{ℓ−1} p_{j}^{s_{j}} ≤ et̃ < ∏_{j=1}^{ℓ−1} p_{j}^{s_{j}+1} ,

and

(10.4c)  δ(g_{s,t̃}u_{y}D^{n+1}) ≤ ce^{−γt} .

Proof. By definition, for some ε > 0 there are infinitely many solutions q̃ to (10.1≤). Therefore, by the above lemma and with γ as in (10.2b), there exists an unbounded set of t ∈ ℝn_{+} such that inequalities (10.2cde) hold for some t_{0} ∈ ℝ and s ∈ ℤℓ−1_{+}. The rest of the proof of Corollary 9.2 applies verbatim. □

10.3 Corollary. Let X be a Besicovitch metric space and μ a uniformly Federer measure on X. Suppose we are given a continuous map f : X → ℚ_{S}^{n} with the follolwing property: for μ-a.e. x_{0} ∈ X there exist a ball B = B(x_{0}, r) and constants C, α, ρ, c_{0} such that conditions (9.6ab) hold for any Δ ∈ 𝔓(D, n+1) and any s ∈ ℤℓ−1_{+}, t ∈ ℤ×ℤn_{+} satisfying (10.4ab). Then f(x) is not VWMA_{≤} for μ-a.e. x ∈ X.

Proof. An application of Theorem 6.3, again with h(x) = g_{s,t̃}u_{f(x)} and m = n + 1, yields

μ (\{x ∈ B | δ(g_{s,t̃}u_{f(x)}D^{n+1}) < ce^{−γt}\}) ≤ (n + 1)C(N_{X}D_{μ}^{2})^{n+1} (\frac{ce^{−γt}}{ρ} )^{α} μ(B)

whenever ce^{−γt} ≤ ρ and (10.4ab) hold. Now observe that for fixed t, the number of different t_{0} ∈ ℤ and s ∈ ℤℓ−1_{+} satisfying (10.4ab) is at most const ·t^{ℓ−1}. Therefore for any c, γ > 0 the sum (over all integers s, t̃ for which inequalities (10.4ab) hold) of measures of sets {x ∈ B | δ(g_{s,t̃}u_{f(x)}D^{n+1}) < ce^{−γt}} converges. As before, an application of the Borel-Cantelli Lemma shows that for μ-a.e. x ∈ B there are at most finitely many integer solutions s, t̃ to (10.4abc) for any c, γ > 0. Corollary 10.2 then implies that f(x) is not VWMA_{≤} for μ-a.e. x ∈ B. □

10.4. Now let us state a lemma showing that y being VWMA_{>} has some implications to the Diophantine properties of its “finite part” (y^{(v)})_{v∈S_{f}}.

Lemma. Assume that y = (y^{(v)})_{v∈S} is VWMA_{>}; then (y^{(v)})_{v∈S_{f}} is VWMA.

Proof. Note that the second inequality in (10.1>) implies

|ỹ^{(∞)} · q̃|_{∞} = |q_{0} + y^{(∞)} · q|_{∞} ≥ |q_{0}| − n∥y^{(∞)}∥_{∞}∥q∥

≥ |q_{0}| − \frac{n∥y^{(∞)}∥_{∞}}{1 + n∥y^{(∞)}∥_{∞}} |q_{0}| = \frac{1}{1 + n∥y^{(∞)}∥_{∞}} |q_{0}| .

Therefore one has

∏_{v∈S_{f}} |ỹ^{(v)} · q̃|_{v} = \frac{c(ỹ · q̃)}{|ỹ^{(∞)} · q̃|_{∞}} ≤ Π_{+}(q)^{−(1+ε)}|q_{0}|^{−ε} · (1 + n∥y^{(∞)}∥_{∞}) |q_{0}|^{−ε}

≤ (1 + n∥y^{(∞)}∥_{∞}) Π_{+}(q)^{−(1+ε)} . □

10.5. Finally we are ready for the

Proof of Theorem 8.7, the case ∞ ∈ S. Applying the case “∞ ∉ S” of Theorem 8.7 to the map (f^{(v)})_{v∈S_{f}}, we obtain that the pushforward of ∏_{v∈S_{f}} μ_{v} by (f^{(v)})_{v∈S_{f}} must be strongly extremal, which, in view of Lemma 10.4, implies that f(x) is not VWMA_{>} for μ-a.e. x ∈ X. Thus, as before, it suffices to take

B = ∏_{v∈S} B_{v} = ∏_{j=1}^{ℓ−1} B_{j} × B_{∞} and μ = ∏_{v∈S} μ_{v} = ∏_{j=1}^{ℓ−1} μ_{j} × μ_{∞} ,

and check that the assumptions of Corollary 10.3 are satisfied by means of writing down explicit expressions for functions x ↦ cov (g_{s,t̃}u_{f(x)}Δ). This again boils down to the computation of components of g_{s,t̃}u_{f(x)}w, where w = ∑_{I⊂{0,...,n}} w_{I}e_{I} ∈ ⋀^{r}(D^{n+1}).

Since there was no change in the ultrametric components of g_{s,t̃} and u_{f(x)}, formula (9.7) is still valid. Furthermore, an expression for the Archimedean components turns out to be similar to (9.7):

(g_{s,t̃}u_{f(x)}e_{I})^{(∞)} = \{ \begin{matrix}e^{−t_{I}} e(∞)_{I} if 0 ∈ I \\ e^{−t_{I}} e(∞)_{I} + ∑_{i∈I} ±e^{−t_{I∪\{0\}∖\{i\}}}f_{i}^{(∞)}(x)e(∞)_{I∪\{0\}∖\{i\}} otherwise . \end{matrix}

Therefore one has (9.8) for j = 1, …, ℓ − 1, and, in addition,

(g_{s,t̃}u_{f(x)}w)^{(∞)} = ∑_{0∉I} w_{I}e^{−t_{I}} e(∞)_{I} + ∑_{0∈I} e^{−t_{I}} (w_{I} + ∑_{i∉I} ±w_{I∪{i}∖{0}}f_{i}^{(∞)}(x))e(∞)_{I} .

We see that real (resp. p_{j}-adic) components of all the coordinates of g_{s,t̃}u_{f(x)}w

are linear combinations of 1, f_{1}^{(∞)}, …, f_{n}^{(∞)} (resp. 1, f_{1}^{(p_{j})}, …, f_{n}^{(p_{j})}). Condition (9.6a) then immediately follows from Lemma 2.1(bc), (8.6) and Corollary 2.3. On the other hand, an argument identical to that of the previous section shows that for every j = 1, …, ℓ−1 there exists ρ_{j} > 0 such that (9.10) holds, and also that there exists ρ_{∞} > 0 such that

sup_{x∈B_{∞} ∩ supp μ_{∞}} ∥(g_{s,t̃}u_{f(x)}w)^{(∞)}∥_{∞} ≥ ρ_{∞} min_{0∈I} e^{−t_{I}} max_{I} |w_{I}|_{∞} ≥ ρ_{∞}e^{−t̃} max_{I} |w_{I}|_{∞} .

Therefore

sup_{x∈B ∩ supp μ} c(g_{s,t̃}u_{f(x)}w) ≥ (∏_{j=1}^{ℓ−1} ρ_{j}p_{j}^{s_{j}} max_{I} |w_{I}|_{p_{j}} ) ρ_{∞}e^{−t̃} max_{I} |w_{I}|_{∞}

≥ ρ_{∞} (∏_{j=1}^{ℓ−1} ρ_{j}) e^{−t̃} (∏_{j=1}^{ℓ−1} p_{j}^{s_{j}} ) max_{I} c(w_{I}) ≥_{(10.4b)} ρ_{∞} ∏_{j=1}^{ℓ−1} \frac{ρ_{j}}{p_{j}} .

This implies (9.6b) and shows that f(x) is not VWMA_{≤} for μ-a.e. x ∈ X, thus finishing the proof of the theorem. □

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Dmitry Kleinbock, Department of Mathematics, Brandeis University, Waltham, MA 02454-9110, USA

   E-mail address: kleinboc@brandeis.edu

George Tomanov, Institut Girard Desargues, Université Claude Bernard – Lyon 1, 69622 Villeurbanne cedex, France

   E-mail address: Georges.Tomanov@desargues.univ-lyon1.fr