ON THE DIMENSION OF SOME LINEAR SPACES OVER FINITE EXTENSION OF ℚ

Part 2.

Leonid A. Gutnik

Think of all the things that will Really be within your reach ! V.Nabokov.

Table of contents

§0. Foreword. §1. Introduction. §2. On the dimension of some linear spaces over a finite extension of the field

ℚ.

§3. Non-discrete free modules over the ring ℤ_{K}. §4. Survey of some results.

§0. Foreword.

The first part of this paper is published here as ’On the rank over ℚ of some real matrices’, Part 1, Max-Plank-Institut für Mathematik, Bonn, Preprint Series, 2002, vol. 27, pages 1 – 32. When I wrote the Part 2, I have decided to change the name of the paper. The paragraph 1 is introductory. In the paragraph 2 I have collected a several working theorems for an estimation of dimension of some linear spaces over a finite extension of ℚ. In the paragraph 3 I introduce and study the notion of measure of nondiscreteness for some free modules over the ring of integers of a finite extension of the field ℚ. A survey of some results which are obtained by means of this method is given in the paragraph 4.

Acknowledgments. I express my deepest thanks for help and support of my work to Professors

B.Z.Moroz, I.I.Piatetski-Shapiro, A.G.Aleksandrov, P.Bundshuh, and S.G.Gindikin.

Typeset by AMS-TEX

§1. Introduction.

We consider a finite extension K/ℚ of the field ℚ, and let

[K : ℚ] = d.

Let K has r_{1} real places and r_{2} complex places. Each such place is the monomorphism of the field K in the field ℝ, if a place is real, or in the field ℂ, if a place is not real; we will denote these monomorphisms respectively by σ_{1} , … σ_{r_{1}+r_{2}}. Then d = r_{1} + 2r_{2}. Let 𝔅 is the fixed integer basis

(2.1.1)  ω_{1} , …, ω_{d}

of the field K over the field ℚ. With extension of the base field of the algebra K/ℚ from ℚ to ℝ appears an isomorphism of the algebra 𝔎 = K ⊗ ℝ onto direct sum

\underbrace{ℝ ⊕ … ⊕ ℝ}_{r_{1} times} ⊕ \underbrace{ℂ ⊕ … ⊕ ℂ}_{r_{2} times}

of r_{1} copies of the field ℝ and of r_{2} copies of the field ℂ.

We will identify the algebra 𝔎 with the specified direct sum by means of this isomorphism. Let π_{j}, where j = 1 , …, r_{1} + r_{2}, denotes projection of the algebra 𝔎 onto j−th among its direct summands and let also π_{j} denotes the natural extension of this projection onto set of all the matrices having all elements in 𝔎. So, π_{j}(𝔎) = ℝ, if j = 1 , …, r_{1} and π_{j}(𝔎) = ℂ, if j = r_{1} + 1 , …, r_{1} + r_{2}. Let further 𝔦_{𝔎} denotes a embedding of the field ℝ by the diagonal way in 𝔎 and let also 𝔦_{𝔎} denotes the extension of this embedding onto set of all the real matrices . So, ℝ is imbedded by means of 𝔦_{𝔎} by diagonal way in 𝔎. Each element Z ∈ 𝔎 has a unique representation in the form:

(2.1.2)  Z = ( z_{1} ; ⋮ ; z_{r_{1}+r_{2}} ; \overline{z_{r_{1}+1}} ; ⋮ ; \overline{z_{r_{1}+r_{2}}} ) ,

where z_{j} = π_{j}(Z) ∈ ℝ for j = 1 , …, r_{1} and z_{j} = π_{j}(Z) ∈ ℂ for j = r_{1}+1 , …, r_{1}+ r_{2}. Let Tr_{𝔎} (Z) denotes the sum

∑_{j=1}^{r_{1}} z_{j} + ∑_{j=r_{1}+1}^{r_{1}+r_{2}} 2 Re (z_{j}) =

∑_{j=1}^{r_{1}} π_{j}(Z) + ∑_{j=r_{1}+1}^{r_{1}+r_{2}} 2 Re (π_{j}(Z),

and let q_{∞}^{(𝔎)}(Z) denotes the value

max(|z_{1}| , …, |z_{r_{1}+r_{2}}|) =
max(|π_{1}(Z)| , …, |π_{r_{1}+r_{2}}(Z)|).

Clearly,

q_{∞}^{(𝔎)}(Z_{1}Z_{2}) ≤ q_{∞}^{(𝔎)}(Z_{1})q_{∞}^{(𝔎)}(Z_{2}),
q_{∞}^{(𝔎)}(Z_{1} + Z_{2}) ≤ q_{∞}^{(𝔎)}(Z_{1}) + q_{∞}^{(𝔎)}(Z_{2}),
q_{∞}^{(𝔎)}(𝔦_{𝔎}(λ)Z) = |λ|q_{∞}^{(𝔎)}(Z)

for any Z_{1} ∈ 𝔎, Z_{2} ∈ 𝔎, Z ∈ 𝔎 and λ ∈ ℝ. Let q_{∞}^{(𝔎)} denotes the natural extension of the specified norm q_{∞}^{(𝔎)} onto set of all the matrices having all elements in 𝔎 (i.e. as the maximum of the norm q_{∞}^{(𝔎)} of the elements of the matrix) . If

Z = ( z_{1} ; ⋮ ; z_{d} ) ∈ K,

then

z_{j} = σ_{j}(Z),

where j = 1 , …, r_{1} + r_{2},

z_{r_{1}+r_{2}+j} = \overline{σ_{r_{1}+j}(Z)},

where j = 1 , …, r_{2}. In particular,

(2.1.3)  ω_{k} = ( σ_{1}(ω_{k}) ; ⋮ ; σ_{r_{1}+r_{2}}(ω_{k}) ; \overline{σ_{r_{1}+1}(ω_{k})} ; ⋮ ; \overline{σ_{r_{1}+r_{2}}(ω_{k})} ) ,

As usually, ℤ_{K} denotes the ring of all the integers of the field K. The ring ℤ_{K} is embedded in the ring 𝔎 as discrete lattice. Moreover, if Z ∈ ℤ_{K}\{0}, then

(∏_{i=1}^{r_{1}} |σ_{j}(Z)|) ∏_{i=1}^{r_{2}} |σ_{r_{1}+i}(Z)|^{2} = | Nm_{K/ℚ}(Z)| ∈ ℕ

and therefore

(2.1.4)  q_{∞}^{(𝔎)}(Z) ≥ 1.

for any Z ∈ ℤ_{K}\{0}. We reserve the name K-integer for denoting of the elements of the ring Z_{K}.

On 𝔎 there is a involution transforming each element (2.1.2) in 𝔎 into

\overline{Z} = ( \overline{z_{1}} ; ⋮ ; \overline{z_{d}} ) ∈ 𝔎.

Every element Z of the algebra 𝔎 has a unique representation in the form

(2.1.6)  Z = ∑_{j=1}^{d} x_{j}ω_{j}

with x_{j} ∈ ℝ and ω_{j} in (2.1.1). Then

K = {∑_{j=1}^{d} x_{j}ω_{j}: x_{j} ∈ ℚ, j = 1 , …, d} ,

ℤ_{K} = {∑_{j=1}^{d} x_{j}ω_{j}: x_{j} ∈ ℤ, j = 1 , …, d} .

For each element Z ∈ 𝔎, which is represented in the form (2.1.6), let q_{∞}^{(𝔅)}(Z) denotes the number

max(|x_{1}| , …, |x_{s}|),

let [Z]_{𝔅} denotes the sum

(2.1.7)  ∑_{j=1}^{d}[x_{j}]ω_{j},

and, finally,let X_{Z}^{(𝔅)} denotes the column

( x_{1} ; ⋮ ; x_{d} ) .

Clearly,

(2.1.8)  [Z]_{𝔅} ∈ ℤ_{K}.

Let

{Z}_{𝔅} = Z − [Z]_{𝔅}

for any Z ∈ 𝔎. Let q_{∞}^{(𝔅)} also denotes the natural extension of the specified norm q_{∞}^{(𝔅)} onto set of all the matrices, which have all the elements in 𝔎 (i.e. as the maximum of the norm q_{∞}^{(𝔅)} of elements of the matrix). So,

(2.1.9)  q_{∞}^{(𝔅)}(Z) = q_{∞}^{(ℝ)}(X_{Z}^{(𝔅)}),

for Z ∈ 𝔎. Clearly, q_{∞}^{(𝔅)}({Z}_{𝔅}) < 1 for any Z ∈ 𝔎. On a finite-dimensional linear space over the field ℝ any two norms are equivalent. Therefore there are constants C_{1} > 0 and C_{2} > 0 such that

(2.1.10)  q_{∞}^{(𝔎)}(Z) ≤ C_{1}q_{∞}^{(𝔅)}(Z)

and

(2.1.11)  q_{∞}^{(𝔅)}(Z) ≤ C_{2}q_{∞}^{(𝔎)}(Z)

for each element Z ∈ 𝔎. Let e_{j}^{∨}, where j = 1 , …, d denotes the j−th column of a unit d × d−matrix.

Let further

e_{k}^{∧} = e_{k}^{∨},

if k = 1 , …, r_{1}, and

e_{k}^{∧} = e_{k}^{∨} + e_{k+r_{2}}^{∨},

if k = r_{1} + 1 , …, r_{1} + r_{2}. Then

(2.1.12)  1_{𝔎} = ∑_{k=1}^{r_{1}+r_{2}} e_{k}^{∧},

(2.1.13)  e_{k}^{∧}e_{k}^{∧} = e_{k}^{∧}, e_{k}^{∧}e_{l}^{∧} = 0,

where k ∈ {1 , …, r_{1} + r_{2}} and l ∈ {1 , …, r_{1} + r_{2}}\{k}.

Let 𝔐 is the unitary 𝔎-module. The idempotent e_{k}^{∧}, where k = 1 , …, r_{1} + r_{2}, determines a 𝔎-submodule e_{k}^{∧}𝔐 of the module 𝔐. According to (2.1.13), the equality (2.1.12) determines splitting of the module 𝔐 into direct sum

e_{1}^{∧}𝔐 ⊕ … ⊕ e_{r_{1}+r_{2}}^{∧}𝔐.

Let ψ_{j}^{(𝔐)}, where j = 1 , …, r_{1} + r_{2}, is the projection of the module 𝔐 onto its direct summand e_{j}M. Then

(2.1.14)  f = ∑_{j=1}^{r_{1}+r_{2}} ψ_{j}^{(𝔐)}(f)

for any f ∈ 𝔐.

Clearly, the unitary 𝔎-module 𝔐 is a free 𝔎-module with finite set of free 𝔎generators if and only if there exists a finite set I and for each k = 1 , …, r_{1} + r_{2} there exists an injection α_{k} of the set I into e_{k}^{∧}𝔐 with the following property:

α_{k}(I) is a basis of the linear space e_{k}^{∧}𝔐 over the field π_{k}(𝔎).

Let 𝔐_{0} denotes the free unitary 𝔎-module with s free 𝔎-generators.

Let

(2.1.15)  e_{1} , …, e_{s},

are free generators of 𝔐_{0} over the ring 𝔎. Let, as above, d = [K : ℚ] and (2.1.1) is the integer basis of K. Let M_{0} is the ℤ_{K}-module

(2.1.16)  M_{0} = {u_{1}e_{1} + … u_{s}e_{s}: u_{1} ∈ ℤ_{K} , …, u_{s} ∈ ℤ_{K}}.

Let, as above, 𝔦_{𝔎} is the embedding of ℝ into 𝔎 by diagonal way. Let 𝔏_{0} = 𝔦_{𝔎}(ℝ)e_{1} + … + 𝔦_{𝔎}(ℝ)e_{s}. We identify the module 𝔐_{0} with a tensor product 𝔎 ⊗_{ℝ} 𝔏_{0}. Then the system

(2.1.17)  ω_{k} ⊗_{ℝ} e_{l} = ω_{k}e_{l}

with k = 1 , …, d and l = 1 , …, d is a basis of 𝔐_{0} over the field ℝ and is a basis of M_{0} over the ring ℤ. Therefore we identify 𝔐_{0} with

(2.1.18)  (𝔎)^{s} = ⊕\begin{matrix}r \\ k=1 \end{matrix}𝔎

and M_{0} with

(ℤ_{K})^{s} = ⊕\begin{matrix}s \\ k=1 \end{matrix}ℤ_{K}.

Therefore every 𝔎-homomorphism ϕ of module 𝔐_{0} into 𝔎-module 𝔐 is uniquely defined by s 𝔎-homomorphisms ϕ_{k} with k = 1 , …, s of the ring 𝔎 into 𝔐 such that ϕ_{k} coincides with ϕ on the k−th direct summand in (2.1.18) and is equal to 0 on others.

Let τ_{k}, where k = 1 , …, r_{1}, denotes the embedding of the field ℝ into ring 𝔎 with the following property:

τ_{k}(α) = αe_{k}^{∨} for any α ∈ ℝ.

Let further τ_{k}, where k = r_{1} + 1 , …, r_{1} + r_{2}, denotes the embedding of ℂ into 𝔎 with the following property:

τ_{k}(α) = αe_{k}^{∨} + \overline{α}e_{k+r_{2}}^{∨} for any α ∈ ℂ.

Then

τ_{k}(α)1_{𝔎} = τ_{k}(α)e_{k}^{∧} = τ_{k}(α),

with α ∈ ℝ, if k = 1 , …, r_{1} and with α ∈ ℂ, if k = r_{1} +1 , …, r_{1} +r_{2}. The embedding τ_{k} defines on the module e_{k}^{∧}𝔐 the structure of linear space over π_{k}(𝔎) = ℝ (respectively π_{k}(𝔎) = ℂ) such that for any g ∈ e_{k}^{∧}𝔐 and for any α ∈ ℝ (respectively α ∈ ℂ) the result of the multiplication of the element g by the number α is equal to τ_{k}(α)g. Clearly,

(2.1.19)  π_{j}(τ_{k}(α)) = 0,

if

{1 , …, r_{1} + r_{2}} ∋ j ≠ k ∈ {1 , …, r_{1} + r_{2}},

but

(2.1.20)  π_{k}(τ_{k}(α)) = α

with α ∈ ℝ, if k ∈ {1 , …, r_{1}}, and with α ∈ ℂ, if k ∈ {r_{1} + 1 , …, r_{1} + r_{2}}. Therefore

ψ_{k}^{(𝔎)}(Z) = τ_{k}(π_{k}(Z)),

where Z ∈ 𝔎 and k = 1 , …, r_{1} + r_{2}. Consequently,

Z = ∑_{k=1}^{r_{1}+r_{2}} τ_{k}(π_{k}(Z))

for any Z ∈ 𝔎. Every 𝔎-homomorphism ϕ of 𝔐 into 𝔎 is uniquely defined as sum

(2.1.21)  ∑_{j=1}^{r_{1}+r_{2}} ϕ_{j},

where ϕ_{j} = ϕψ_{j}^{(𝔐)} for any j ∈ {1 , …, r_{1} + r_{2}}; moreover, if β_{j} are an arbitrary 𝔎-homomorphisms of the module e_{j}^{∧}𝔐 into 𝔎 and we take

ϕ_{j}(f) = β_{j}(f), ϕ_{k}(f) = 0

for any j ∈ {1 , …, r_{1}+r_{2}}, for any k ∈ {1 , …, r_{1}+r_{2}}\{j} and for any f ∈ e_{j}^{∧}M,) then (2.1.21) defines 𝔎-homomorphism ϕ of the module 𝔐 into 𝔎 such that ϕ coincides with β_{j} on the module e_{j}^{∧}𝔐.

The 𝔎-homomorphism ϕ_{j} with j = 1 , …, r_{1} is uniquely defined by means of the linear over ℝ functions

(2.1.22)  ϕ_{j,k} = π_{k}ϕ_{j},

where k = 1 , …, r_{1},

(2.1.23)  ϕ_{j,k} = Re (π_{k}ϕ_{j}), ϕ_{j,k+r_{2}} = Im (π_{k}ϕ_{j}),

where k = r_{1} + 1 , …, r_{1} + r_{2}, such that

ϕ_{j}(e_{j}^{∧}f) = (∑_{k=1}^{r_{1}} τ_{k}(ϕ_{j,k}(e_{j}^{∧}f))e_{k}^{∧}) +

( ∑_{k=r_{1}+1}^{r_{1}+r_{2}} τ_{k}(ϕ_{j,k}(e_{j}^{∧}f) + iϕ_{j,k+r_{2}}(e_{j}^{∧}f))) e_{k}^{∧}.

The 𝔎-homomorphism ϕ_{j} with j = r_{1} +1 , …, r_{1} +r_{2} is uniquely defined by means of linear over ℝ functions

(2.1.24)  ϕ_{j,k} = Re (π_{k}ϕ_{j}),

where k = 1 , …, r_{1} + r_{2},

ϕ_{j}(e_{j}^{∧}f) = (∑_{k=1}^{r_{1}} τ_{k}(ϕ_{j,k}(e_{j}^{∧}f))e_{k}^{∧}) +

( ∑_{k=r_{1}+1}^{r_{1}+r_{2}} τ_{k}(ϕ_{j,k}(e_{j}^{∧}f) − iϕ_{j,k}(τ_{j}(i)e_{j}^{∧}f))) e_{k}^{∧}.

So,

(2.1.25)  ϕ(f) = ∑_{j=1}^{r_{1}+r_{2}} ϕ(π_{j}^{(𝔐)}(f)) =

( ∑_{j=1}^{r_{1}+r_{2}} ∑_{k=1}^{r_{1}} τ_{k}(π_{k}(ϕ(π_{j}^{(𝔐)}(f))))) +

(∑_{j=1}^{r_{1}} ∑_{k=r_{1}+1}^{r_{1}+r_{2}} τ_{k}(π_{k}(ϕ(π_{j}^{(𝔐)}(f))))) +

( ∑_{j=r_{1}+1}^{r_{1}+r_{2}} ∑_{k=r_{1}+1}^{r_{1}+r_{2}} τ_{k}(π_{k}(ϕ(π_{j}^{(𝔐)}(f))))) =

( ∑_{j=1}^{r_{1}+r_{2}} ∑_{k=1}^{r_{1}} τ_{k}(ϕ_{j,k}(f))) +

(∑_{j=1}^{r_{1}} ∑_{k=r_{1}+1}^{r_{1}+r_{2}} τ_{k}(ϕ_{j,k}(f) + iϕ_{j,k+r_{2}}(f))) +

( ∑_{j=r_{1}+1}^{r_{1}+r_{2}} ∑_{k=r_{1}+1}^{r_{1}+r_{2}} τ_{k}(ϕ_{j,k}(f) − iϕ_{j,k}(τ_{j}(i)f))) =

( ∑_{j=1}^{r_{1}+r_{2}} ∑_{k=1}^{r_{1}} τ_{k}(ϕ_{j,k}(ψ_{j}^{(𝔐)}(f)))) +

(∑_{j=1}^{r_{1}} ∑_{k=r_{1}+1}^{r_{1}+r_{2}} τ_{k}(ϕ_{j,k}(ψ_{j}^{(𝔐)}(f)) + iϕ_{j,k+r_{2}}(ψ_{j}^{(𝔐)}(f)))) +

( ∑_{j=r_{1}+1}^{r_{1}+r_{2}} ∑_{k=r_{1}+1}^{r_{1}+r_{2}} τ_{k}(ϕ_{j,k}(ψ_{j}^{(𝔐)}(f)) − iϕ_{j,k}(τ_{j}(i)ψ_{j}^{(𝔐)}(f))))

for any f ∈ 𝔐.

Clearly, the elements

(2.1.26)  g_{1} ∈ 𝔐 , …, g_{s} ∈ 𝔐

compose a linearly independent system over over K if and only if the elements

(2.1.27)  ω_{k}g_{l}

with k = 1 , …, d, l = 1 , …, s and with basis (2.1.1) of the field K compose a linearly independent system over ℚ.

Definition 2.1.1. Let 𝔐 is a unitary 𝔎-module. We say that the function q on 𝔐 is a norm, when the following conditions are realized:

(i) q(f) ≥ 0 for any f ∈ 𝔐; the equalities q(f) = 0 and f = 0 are equivalent; (ii) the inequality q(f_{1} + f_{2}) ≤ q(f_{1}) + q(f_{2}) (i.e. the inequality of triangle) is

fulfilled for any f_{1} ∈ 𝔐 and any f_{2} ∈ 𝔐; (iii) the equality

q(f) = max_{j∈{1 ,... ,r_{1}+r_{2}}} q(e_{j}^{∧}λf) = q_{∞}^{(𝔎)}(e_{j}^{∧}λ)q(e_{j}f)

holds for any f ∈ 𝔐 and for any j ∈ {1 , …, r_{1} + r_{2}};

(iv) the equality q(e_{j}^{∧}(λf)) = |π_{j}(λ)|q(e_{j}f) is fulfilled for any f ∈ 𝔐, for any λ ∈ 𝔎 and for any j ∈ {1 , …, r_{1} + r_{2}}.

According to (iii) and (iiii), if λ ∈ ℝ and f ∈ 𝔐, then q(𝔦_{𝔎}(λ)f) = |λ|q(f). So, 𝔐 is the real linear normed space relative to the norm q.

Let 𝔐 is a normed 𝔎-module. Let ϕ is a 𝔎-homomorphism of the module 𝔐 into 𝔎. Clearly, the homomorphism ϕ is continuous if and only if the linear functionals (2.1.22),(2.1.23) and (2.1.24) are continuous. Let

(2.1.28)  q_{∞}^{∼}(ϕ) = sup_{f∈𝔐,q(f)=1} q_{∞}^{𝔎}(ϕ(f)).

(2.1.29)  q_{0}^{∼}(ϕ) = ∑_{j=1}^{r_{1}} ∑_{k=1}^{d} q_{∞}^{∼}(ϕ_{j,k}) + ∑_{j=r_{1}+1}^{r_{1}+r_{2}} ∑_{k=1}^{r_{1}+r_{2}} q_{∞}^{∼}(ϕ_{j,k}).

Clearly,

q_{∞}^{∼}(ϕ) ≤ q_{0}^{∼}(ϕ) ≤ d^{2}q_{∞}^{∼}(ϕ).

Therefore the 𝔎-homomorphism ϕ of the normed 𝔎-module into 𝔎 is continuous if and only if

(2.1.30)  q_{0}^{∼}(ϕ) < ∞;

if the inequality (2.1.30) is fulfilled, we say that ϕ is bounded. So, we have the following analog of Hahn-Banach Theorem:

Lemma 2.1.8.Let 𝔐 is a normed 𝔎-module and let 𝔐_{1} is 𝔎−submodule of the module 𝔐. Let further ϕ^{∨} is a continuous 𝔎-homomorphism of the module 𝔐_{1} into 𝔎.

Then there exists the continuous 𝔎−-homomorphism ϕ of the module 𝔐 into 𝔎 such that ϕ coincides with ϕ^{∨} on 𝔐_{1} and

q_{0}^{∼}(ϕ) = q_{0}^{∼}(ϕ^{∨}).

Proof. The assertion of the Lemma is the direct corollary of equalities (2.1.25), (2.1.29) and of the Hahn-Banach Theorem (more precisely, of real and complex form of this Theorem). ■

We shall continue the norm q_{∞}^{(𝔎)} by the usual way on the 𝔎-module 𝔐_{0}, if we shall put

q_{∞}^{(𝔎)}(f) = max(q_{∞}^{(𝔎)}(a_{1}) , …, q_{∞}^{(𝔎)}(a_{s}))

for any

f = a_{1}e_{1} + … + a_{s}e_{s} ∈ 𝔐_{0}

with a_{1} ∈ 𝔎 , …, a_{s} ∈ 𝔎. If this continuation of the norm q_{∞}^{(𝔎)} on 𝔐_{0} will be done, then each 𝔎− homomorphism of module 𝔐_{0} into 𝔎 becomes continuous.

§2. On the dimension of some linear spaces over finite extension of the field ℚ.

Let, as above, 𝔦_{𝔎} denotes the embedding of ℝ in 𝔎 by the diagonal way and also denotes the extension of this embedding onto the set of all the real matrices .

Definition 2.2.1. Let A = (a_{i,k}) is the m × n-matrix with a_{i,k} ∈ 𝔎, where i = 1 , …, m, k = 1 , …, n. Let further E_{m} denotes the unit m × m-matrix. We say that the matrix A is K-quasi-irrational, if the system of columns of the matrix

(2.2.1)  (A, 𝔦_{𝔎}(E_{m}))

is linearly independent over K.

The following examples illustrate this definition:

if [K : ℚ] = 2, then the column ( \begin{matrix}1 \\ 0 \end{matrix} ) ∈ 𝔎 is K-quasi-irrational. In fact, among

elements of K only 𝔦_{𝔎}(1) = 1_{𝔎} = ( \begin{matrix}1 \\ 1 \end{matrix} ) has the first coordinate equal to 1; on the

other hand the column ( \begin{matrix}𝔦_{𝔎}(1) \\ 𝔦_{𝔎}(0) \end{matrix} ) = ( \begin{matrix}( & {1 \atop 1} & ) \\ ( & {0 \atop 0} & ) \end{matrix} ) is not K-quasi-irrational. For the

field K = ℚ[\sqrt{2}], the column ( \begin{matrix}√ & \overline{2} \\ −\sqrt{2} \end{matrix} ) ∈ 𝔎 is quasi-irrational but not K-quasi-

irrational; on the other hand the column ( \begin{matrix}√ & \overline{2} \\ √ & \overline{2} \end{matrix} ) ∈ 𝔎 is K-quasi-irrational.

The following simple Lemma specifies the domain of applications and of interest of definition 2.2.1.

Lemma 2.2.1. Let A is the real m × n-matrix, let B = (A, E_{m}), let

B_{1} = (ω_{1} ⊗ B , …, ω_{r_{1}} ⊗ B),

if r_{1} > 0, let

B_{2} = (Re (ω_{r_{1}+1}) ⊗ B, Im (ω_{r_{1}+1}) ⊗ B , …, Re (ω_{r_{1}+r_{2}}) ⊗ B, Im (ω_{r_{1}+r_{2}}) ⊗ A), if r_{2} > 0, let B_{3} = B_{1}, if r_{2} = 0, let B_{3} = B_{2}, if r_{1} = 0 and, finally, let B_{3} = (B_{1}, B_{2}), if r_{1}r_{2} > 0.

Then the matrix 𝔦_{𝔎}(A) is K-quasi-irrational if and only if the system of columns of matrix B_{3} is linearly independent over ℚ. In particular, if the matrix 𝔦_{𝔎}(A) is K-quasi-irrational, then the matrix A is quasi-irrational.

Proof.The assertion of the lemma is trivial. ■

Lemma 2.2.2. Let p ∈ ℕ, q ∈ p + ℕ. Let further a_{i,k} ∈ 𝔎 for i = 1 , …, p and k = 1 , …, q. Let, finally,

L_{i} = a_{i,1}z_{1} + … + a_{i,q}z_{q},

where i = 1 , …, p.

Then there exists a constant C > 0 and, for each ν ∈ ℕ, there are Z_{1} ∈ ℤ_{K} , …, Z_{q} ∈ ℤ_{K}, which satisfy to the following inequalities :

q_{∞}^{(𝔎)}(Z_{1}) + … + q_{∞}^{𝔎}(Z_{q}) > 0,

q_{∞}^{(𝔎)}(a_{i,1}Z_{1} + … + a_{i,q}Z_{q}) ≤ \frac{1}{ν}

for i = 1, ..., p and

q_{∞}^{(𝔎)}(Z_{k}) < Cν^{\frac{p}{q−p}}

for k = 1, ..., q.

Proof. If K = ℚ, then 𝔎 = ℝ and the Lemma coincides with well known Siegel Lemma. Representing Z_{k} in the form

Z_{k} = ∑_{l=1} ω_{l}x_{l,k},

where ω_{l} with l = 1 , …, d is the basis (2.1.1), we reduce the Lemma to the case of dp linear forms with real coefficients depending from dq variables. We obtain the assertion of the Lemma 2.2.2, applying Siegel’s Lemma now . ■

Lemma 2.2.3. The following assertions are equivalent:

(i) if X ∈ K^{n}\{0}, then AX ∉ K^{m};

(ii) there are

(2.2.2)  α_{l}(ν) ∈ ℤ_{K},

where ν ∈ ℕ, l = 1 … m + n, which satisfy to the condition

(2.2.3)  lim_{ν→∞} q_{∞}^{(𝔎)} (−α_{m+k}(ν) + ∑_{j=1}^{m} a_{j,k}α_{j}(ν)) = 0,

where k = 1 , …, n and, for each κ ∈ ℕ, the system, composed by the infinite sequences

(2.2.4)  α_{l}(κ + 1) , …, α_{l}(κ + μ) , …,

where μ runs through ℕ and l = 1 , …, m + n, is linearly independent system over 𝕂

(iii) the matrix A is K-quasi-irrational.

Moreover if the assertion (iii) is true, then α_{l}(ν) may be chosen in such a way that the following conditions will be fulfilled:

q_{∞}^{(𝔎)}(α_{l}(ν)) = O(ν^{n}),

where ν ∈ ℕ, l = 1 , …, m + n, and

q_{∞}^{(𝔎)} (−α_{m+k}(ν) + ∑_{j=1}^{m} a_{j,k}α_{j}(ν)) = O(1/ν),

where ν ∈ ℕ, k = 1 , …, n.

Proof. Clearly,condition (i) is equivalent to (iii).

Let the assertion (ii) is true. Then there exists a column X such that X ∈ K^{n}\{0} and AX ∈ K^{m}. On choosing a suitable d ∈ ℕ and replacing of column X by the column dX, if necessary, we may assume that

X ∈ ℤ_{K}^{n}\{0} , AX ∈ ℤ_{K}^{m}.

Let x_{k}, for k = 1 , …, n,is the k−th element of the column X. According to (2.2.3),

(2.2.5)  q_{∞}^{(𝔎)} ((− ∑_{k=1}^{n} x_{k}α_{m+k}(ν)) + ∑_{j=1}^{m} α_{j}(ν) ∑_{k=1}^{n} a_{j,k}x_{k}) =

q_{∞}^{(𝔎)} (∑_{k=1}^{n} x_{k} (−α_{m+k}(ν) + ∑_{j=1}^{m} a_{j,k}α_{j}(ν))) ≤

∑_{k=1}^{n} q_{∞}^{(𝔎)}(x_{k})q_{∞}^{(𝔎)} (−α_{m+k}(ν) + ∑_{j=1}^{m} α_{j}(ν)a_{j,k}) .

But

lim_{ν→∞} (∑_{k=1}^{n} q_{∞}^{(𝔎)}(x_{k})q_{∞}^{(𝔎)} (−α_{m+k}(ν) + ∑_{j=1}^{m} α_{j}(ν)a_{j,k})) = 0.

Therefore

lim_{ν→∞} (q_{∞}^{(𝔎)} ((− ∑_{k=1}^{n} x_{k}α_{m+k}(ν)) + ∑_{j=1}^{m} α_{j}(ν) ∑_{k=1}^{n} a_{j,k}x_{k})) = 0.

Consequently, there exists a number κ ∈ ℕ with the following property: for any μ ∈ ℕ, the value of the norm q_{∞}^{(𝔎)} of the element

(2.2.6)  (− ∑_{k=1}^{n} x_{k}α_{m+k}(κ + μ)) + ∑_{j=1}^{m} α_{j}(κ + μ) ∑_{k=1}^{n} a_{j,k}x_{k} ∈ ℤ_{K}

is less than 1; therefore, according to (2.1.4), this element of the ring ℤ_{K} is equal to 0; however this is impossible for every μ in ℕ because the sequences (2.2.4) compose a linearly independent system over K and x_{k} ≠ 0 for some k = 1 , …, n.

Let the assertion (iii) is true. For each i ∈ {1 , …, m}, we consider a system of n linear forms

(2.2.7)  L_{i,k} = y_{0}a_{i,k} − y_{k}, k = 1 , …, n,

which depend from variables y_{0} , …, y_{n}.

Since the number n+1 of the variables exceeds the number n of the linear forms, in view of the Lemma 2.2.2, there exists a constant C > 0 which has the following properties:

for each μ ∈ ℕ and for each k = 1 , …, n there are elements θ_{k}(i+m(μ−1)) ∈ 𝔎 and elements y_{0} , …, y_{n} in the ring ℤ_{K}, which satisfy to the conditions

(2.2.8)  q_{∞}^{(𝔎)}(θ_{k}(i + m(μ − 1)) ≤ 1,
q_{∞}^{(𝔎)}(y_{0}) ≤ C(i + m(μ − 1))^{n} , …, q_{∞}^{(𝔎)}(y_{n}) ≤ C(i + m(μ − 1))^{n}

L_{i,k} = \frac{1}{i + m(μ − 1)}θ_{k}(i + m(μ − 1),

and y_{i} ≠ 0 for some i; therefore y_{0} ≠ 0. For these values of the variables y_{0} , …, y_{n}, we take

α_{i}(i + m(μ − 1)) = y_{0}, α_{m+k}(i + m(μ − 1)) = y_{k},

with k = 1 , …, n, and

α_{l}(i + m(μ − 1)) = 0

for l ∈ {1 , …, m}\{i}. We have thus constructed sequences of elements of the ring ℤ_{K}

α_{l}(ν), ν ∈ ℕ,

with l = 1 , …, m + n, which satisfy to the following conditions:

α_{i}(ν) ≠ 0

for ν ≡ i (mod m) and i ∈ {1 , …, m},

α_{j}(ν) = 0

for ν ≡ i (mod m) and j ∈ {1 , …, m}\{i},

q_{∞}^{(𝔎)}(α_{l}(ν)) ≤ Cν^{n}

and

(2.2.9)  −α_{m+k}(ν) + ∑_{j=1}^{m} a_{j,k}α_{j}(ν) =

−α_{m+k}(ν) + a_{i,k}α_{i}(ν) = \frac{θ_{k}(ν)}{ν},

for ν ≡ i (mod m) , k = 1 , …, n. Let for some κ ∈ ℕ the sequences (2.2.4) compose the linearly dependent system over K. Then one can find elements t_{l} ∈ ℤ_{K}, l = 1 , …, m + n, with property

(2.2.10)  ∑_{j=1}^{m} t_{j}α_{j}(κ + μ) + ∑_{k=1}^{n} t_{k+m}α_{m+k}(κ + μ)) = 0

and t_{j} ≠ 0 for some j. According to(2.2.9) and (2.2.10),

∑_{j=1}^{m} t_{j}α_{j}(κ + μ) + ∑_{k=1}^{n} t_{k+m} (∑_{j=1}^{m} a_{j,k}α_{j}(κ + μ)\frac{1}{ν} θ_{k}) = 0,

(2.1.11)  ∑_{j=1}^{m} t_{j}α_{j}(κ + μ) (t_{j} + ∑_{k=1}^{n} t_{k+m}a_{j,k}) = ∑_{k=1}^{n} t_{k+m} \frac{1}{κ + μ}θ_{k}

For i ∈ [1, m] ∩ ℕ, let

(2.2.12)  μ ≡ −κ + i (mod m).

Then α_{i}(κ + μ) ≠ 0, q^{(𝔎)}(α_{i}(κ + μ)) ≥ 1, α_{j}(κ + μ) = 0 for j ≠ i and, in view of (2.2.11),

q^{(𝔎)} (t_{i} + ∑_{k=1}^{n} t_{k+m}a_{i,k}) ≤

(2.2.13)  (κ + μ)^{−1} ∑_{k=1}^{n} q^{(𝔎)}(t_{k+m}).

On passing to the limit μ → ∞, we obtain from (2.2.12) and (2.2.13) the following equality

t_{i} + ∑_{k=1}^{n} t_{k+m}a_{i,k} = 0,

with i = 1 , …, m. But this contradicts to (iii). ■.

Lemma 2.2.4. Let 𝔐_{1} is a normed 𝔎-module with s 𝔎-generators

(2.2.14)  f_{1} , …, f_{s}.

If 𝔐_{1} is free 𝔎-module with m free generators, then 𝔐_{1} has a system of 𝔎generators

(2.2.15)  g_{1} , …, g_{s}

with following property:

the systems (2.2.14) and (2.2.15) are equivalent over ℚ and subsystem, consisting of last m elements of system (2.2.15) is the system of free 𝔎 − generators of the module 𝔐_{1}.

Proof. For each i = 1 , …, r_{1} + r_{2}, let is chosen a basis of the system

(2.2.16)  e_{i}^{∧}f_{1} , …, e_{i}^{∧}f_{s},

where, as above, e_{i}^{∧} = π_{i}(1_{𝔎}) and A_{i} is the m × s − matrix, composed from columns of coordinates of elements (2.2.16) in this basis. Let further X is the s×m − matrix, consisting of sm independent variables x_{k,l}, where k = 1 , …, s and l = 1 , … m. Let further P_{i}(X) = det(AX). For each i = 1 , …, r_{1} + r_{2}, there is a specialization X = X_{i} ∈ (π_{i}(𝔎))^{sm} with property P_{i}(X_{i}) ≠ 0. Therefore P_{i}(X) ≠ 0 and P_{0}(X) = ∏_{i=1}^{m} P_{i}(X) ≠ 0. Therefore there is specialization X = X_{0} ∈ (ℚ)^{sm} with property P_{0}(X_{0}) ≠ 0. Clearly, the columns of the s×m − matrix X_{0} compose a linear independent system over ℚ. But s ≥ m. Therefore there exists a matrix Y ∈ Mat_{s}(ℚ), with the matrix X on the last columns. ■

Theorem 2.2.1 Let 𝔐 is the normed 𝔎-module and 𝔐_{0} is the free 𝔎-module with free generators (2.1.15). Let (2.2.14) are some elements of 𝔐. In order that the elements (2.2.14) compose a linearly independent system over K it is sufficient the existence of 𝔎-homomorphism α_{0} of the module 𝔐_{0} into 𝔐,

the existence,for ν ∈ ℕ, of continuous 𝔎-homomorphism ϕ_{ν}^{∨} of 𝔎-module 𝔐 into 𝔎

and existence, for ν ∈ ℕ, of 𝔎-homomorphism ψ_{ν} of 𝔎-module 𝔐_{0} into 𝔎 with the following properties:

(a) the system composed by the elements

(2.2.17)  α_{0}(e_{1}) , …, α_{0}(e_{s})

is a system equivalent over K to the system (2.2.14); (b)

(2.2.18)  ψ_{ν}(f) ∈ ℤ_{K}

for any f ∈ M_{0} and ν ∈ ℕ; (c)

(2.2.19)  lim_{ν→∞} q_{∞}^{(𝔎)}(ϕ_{ν}(α_{0}(f)) − ψ_{ν}(f)) = 0

for any f ∈ 𝔐_{0}; (d)

(2.2.20)  ⋂_{ν=μ}^{∞} ker(ψ_{ν}) = {0}

for any μ ∈ ℕ.

If 𝔎-module 𝔐_{1} generated by system (2.2.14) is a free 𝔎-module, then the conditions specified above are also necessary. Moreover, if 𝔐_{1} is a free 𝔎-module with m generators, then the homomorphisms α_{0}, ϕ_{ν} and ψ_{ν} may be chosen such that

(2.2.21)  q_{∞}^{∼}(ϕ_{ν}) = O(ν^{s−m}), q_{∞}^{∼}(ψ_{ν}) = O(ν^{s−m)}), q_{∞}^{∼}(ϕ_{ν} − ψ_{ν}) = O(ν^{−1}).

Proof.

Sufficiency. Let the equality

(2.2.22)  u_{1}α_{0}(e_{1}) + … u_{s}α_{0}(e_{s}) = 0.

is fulfilled for some elements

u_{1} ∈ ℤ_{K} , …, u_{s} ∈ ℤ_{K}.

In view of (2.1.16),

f = u_{1}e_{1} + … + u_{s}e_{s} ∈ M_{0}

and, according to (2.2.18),

(2.2.23)  ψ_{ν}(f) ∈ ℤ_{K},

where ν ∈ ℕ. In view of (2.2.22),

(2.2.24)  α_{0}(f) = 0.

In view of (2.2.19) and (2.2.22),

(2.2.25)  lim_{ν→∞}(q_{∞}^{(𝔎)}(ψ_{ν}(f))) = 0.

Therefore there exists μ ∈ ℕ with the following property:

(2.2.26)  q_{∞}^{(𝔎)}(ψ_{ν}(f)) < 1

for all the integers ν ∈ μ − 1 + ℕ. According to (2.1.4) and (2.2.24),

(2.2.26)  ψ_{ν}(f) = 0

for all the integers ν ∈ μ − 1 + ℕ. According to (2.2.26) and (2.2.20), f = 0.

Necessity. According to the Lemma 2.2.4, the module 𝔐_{1} has a system of 𝔎-generators 2.2.15 with following property:

the systems (2.2.14) and (2.2.15) are equivalent over ℚ and subsystem,consisting of last m elements of system (2.2.15) is the system of free 𝔎-generators of the module 𝔐_{1}.

Since the elements (2.2.14) are linearly independent over K, the elements (2.2.15) are also linearly independent over K. Let

(2.2.27)  n = s − m.

Then the system

(2.2.28)  g_{n+k},

where k = 1 , …, m, is a system of free generators of 𝔐_{1}. Clearly, there exists a unique 𝔎-homomorphism α_{0} of 𝔐_{0} into the module 𝔐_{1} with following property:

(2.2.29)  α_{0}(e_{l}) = g_{l},

where l = 1 , …, s.

Clearly, for any j = 1 , …, m, there exists a unique 𝔎-homomorphism θ_{j} of the 𝔎-module 𝔐_{1} into 𝔎 with the following property:

θ_{j}(g_{n+k}) = 0_{𝔎} for any k ∈ {1 , …, m}\{j} and θ_{k}(g_{n+k}) = 1_{𝔎} for each k ∈ {1 , …, m}.

Also for j = 1 , …, s, there exists a unique 𝔎-homomorphism η_{j} of the 𝔎 − module 𝔐_{0} into 𝔎 with the following property:

η_{j}^{∨}(e_{k}) = 0_{𝔎} for k ∈ {1 , …, s}\{j} and η_{k}(e_{k}) = 1_{𝔎} for k ∈ {1 , …, s}.

Then η_{k} = θ_{k}^{∨}α_{0} for k ∈ {n + 1 , …, s}. Clearly, the homomorphism α_{0} is continuous, the homomorphism η_{k} with k ∈ {1 , …, s} is continuous and homomorphism θ_{k} with k ∈ {1 , …, m} is continuous. According to the lemma 2.1.8, for k ∈ {1 , …, m}, there exists a continuous 𝔎-homomorphism θ_{k}^{∧} of the module 𝔐 into 𝔎 with the following property:

θ_{k}^{∧} coincides with θ_{k} on the module 𝔐_{1}.

Then

η_{k} = θ_{k}^{∧}α_{0}

for k ∈ {n + 1 , …, s}. Clearly, η_{k}(M_{0}) ⊂ ℤ_{K} for k ∈ {1 , …, s}. Clearly,

⋂_{k=1}^{s} ker(η_{k}) = {0_{𝔎}}.

If n = 0, then, for k ∈ {1 , …, s}, ν ∈ ℕ, ν ≡ k mod s, we can put

ϕ_{ν} = θ_{k}^{∧}, ψ_{ν} = η_{k}.

Then (2.2.18), (2.2.19) and (2.2.21) are fulfilled.

If m < s, then according to (2.2.27) and (2.2.28),

(2.2.30)  g_{n+1} , …, g_{n+m}

is system of free 𝔎-generators of 𝔐_{1} and therefore

(2.2.31)  g_{k} = ∑_{i=1}^{m} a_{i,k}g_{n+i},

where a_{i,k} ∈ 𝔎 for k = 1 , …, n, i = 1 , …, m. According to the Lemma 2.2.3, there are

(2.2.32)  α_{l}(ν) ∈ ℤ_{K},

where ν ∈ ℕ, l = 1 … m + n, which satisfy to the following conditions:

(i)

q_{∞}^{(𝔎)}(α_{l}(ν)) = O(ν^{n})

where l = 1 , …, m + n and ν ∈ ℕ; (ii)

q_{∞}^{(𝔎)} (−α_{m+k}(ν) + ∑_{j=1}^{m} a_{j,k}α_{j}(ν)) = O(ν^{−1}),

where k = 1 , …, n and ν ∈ ℕ; (iii) for each κ ∈ ℕ, the system, composed by the infinite sequences

α_{l}(κ + 1) , …, α_{l}(κ + μ) , …,

where μ runs through ℕ and l = 1 , …, m + n, is linearly independent system over the field 𝕂.

Let

(2.2.33)  ψ_{ν} = (∑_{i=1}^{m} α_{i}(ν)η_{n+i}) + ∑_{k=1}^{n} α_{m+k}(ν)η_{k},

ϕ_{ν} = ∑_{i=1}^{m} α_{i}(ν)θ_{n+i}.

Then

(2.2.34)  ϕ_{ν}(α_{0}(e_{n+i})) = ϕ_{ν}(g_{n+i}) = ψ_{ν}(e_{n+i}) = α_{i}(ν) = O(ν^{n}),

where i = 1 , …, m,

(2.2.35)  ψ_{ν}(e_{k}) = α_{m+k}(ν) = O(ν^{n}),

where k = 1 , …, n,

(2.2.36)  ϕ_{ν}(g_{k}) = ϕ_{ν} (∑_{i=1}^{m} a_{i,k}g_{n+i}) = ∑_{i=1}^{m} a_{i,k}ϕ_{ν}(g_{n+i}) =

∑_{i=1}^{m} a_{i,k}α_{i}(ν) = O(ν^{n}),

where k = 1 , …, n,

(2.2.37)  ϕ_{ν}(α_{0}(e_{k})) − ψ_{ν}(e_{k}) = ϕ_{ν}(g_{k}) − ψ_{ν}(e_{k}) =

(∑_{i=1}^{m} a_{i,k}α_{i}(ν)) − α_{m+k}(ν) = O(1/ν).

We see that (2.2.18), (2.2.19) and (2.2.21) follow from

(2.2.33), (2.2.34), (2.2.35),(2.2.36) and (2.2.37) .

Finally, (2.2.20) follows from the (2.2.33), (2.2.32) and (iii) . ■

Let 𝔏 is a linear normed space over the field ℝ and 𝔏′ is the linear space conjugate over the field ℝ to the space 𝔏. Let q denotes the norm on the space 𝔏 and q′ denotes the norm which is defined on the space 𝔏′ and is the norm associated with q. Let 𝔏_{0} is a linear space over the field ℝ. Let further s = dim_{ℝ}(𝔏_{0}) and

(2.2.38)  e_{1} , …, e_{s}

is the basis of the space 𝔏_{0} over the field ℝ. Let

(2.2.39)  L_{0} = {x_{1}e_{1} + … x_{s}e_{s}: x_{1} ∈ ℤ , …, x_{s} ∈ ℤ}.

Let 𝔏′_{0} is the linear space conjugate over the field ℝ to the space 𝔏_{0} and

(2.2.40)  η_{1} , …, η_{s}

is the basis of the space 𝔏′_{0} biorthonormal with the basis (2.2.38). Then

(2.2.41)  η_{i}(x_{1}e_{1} + … x_{s}e_{s}) = x_{i},

where x_{1} ∈ ℝ , …, x_{s} ∈ ℝ and

(2.2.42)  L_{0}^{∧} = {x_{1}η_{1} + … x_{s}η_{s}: x_{1} ∈ ℤ , …, x_{s} ∈ ℤ}.

is the lattice dual to the lattice L_{0}. Let, finally, q_{0} denotes a some norm on 𝔏_{0} and q′_{0} denotes the norm which is defined on 𝔏′_{0} and is the norm associated with q_{0}. Theorem 2.2.2. Let the space 𝔏 contains all the elements of the system 2.2.14. Let further 𝔏_{1} is the linear envelope over the field ℝ of the system 2.2.14. In order that the elements (2.2.14) compose a linearly independent system over the field ℚ it is necessary and sufficient the existence of a linear map α_{0} of the space 𝔏_{0} into 𝔏, the existence of a sequence of a linear over the field ℝ continuous functionals ϕ_{ν}^{∨} ( where ν ∈ ℕ ) on the linear normed space 𝔏 and the existence of sequence of a linear over ℝ functionals ψ_{ν}^{∨} ( where ν ∈ ℕ ) on the linear normed space 𝔏_{0} such that the system

α_{0}(e_{1}) , …, α_{0}(e_{s})

would be equivalent over the field ℚ to the system (2.2.14) and the following conditions would be fulfilled:

ψ_{ν}(f) ∈ ℤ

for any f ∈ L_{0};

(2.2.43)  lim_{ν→∞}(ϕ_{ν}(α_{0}(f)) − ψ_{ν}(f)) = 0

for any f ∈ 𝔏_{0}; and

(2.2.44)  ⋂_{ν=μ}^{∞} Ker (ψ_{ν}) = {0}

for any μ ∈ ℕ. Moreover, if (2.2.43) and (2.2.44) are fulfilled, then

(2.2.45)  dim_{ℚ}(α_{0}(𝔏_{0})) = s;

if the equality (2.2.39) is fulfilled and

m = dim_{ℝ}(𝔏_{1}),

then there exists a linear over ℝ map α_{0} of space 𝔏_{0} into space 𝔏, there exists a sequence of linear functions ϕ_{ν} ∈ 𝔏′ ( where ν ∈ ℕ ) and there exists a sequence of linear functions ψ_{ν} ∈ L_{0}^{∧} ( where ν ∈ ℕ ) such that the equality (2.2.44) holds and the following conditions are fulfilled:

(2.2.46)  q′(ϕ_{ν}) = O(ν^{s−m+1}), q′_{0}(ψ_{ν}) = O(ν^{s−m}),

(2.2.47)  |ϕ_{ν}(α_{0}(f)) − ψ_{ν}(f)| = q_{0}(f)O(1/ν)

for any f ∈ 𝔏_{0}.

Proof. Let K = ℚ. Then 𝔎 = ℝ. Since each ℝ-module is free, it follows that the Theorem 2.2.2 is direct corollary of the Theorem 2.2.1. ■

Corollary 1. Let B = (b_{i,k}), i = 1 , …, p, k = 1 , …, s is a p × s-matrix,all the elements of which are real numbers. In order that the columns of matrix B compose a linearly independent system over the field ℚ it is necessary and sufficient the existence of p sequences of real numbers β_{1}(ν) , …, β_{p}(ν) and the existence of s sequences of integer numbers ξ_{1}(ν) , …, ξ_{s}(ν) such that the following conditions are fulfilled:

(I)

lim_{ν→∞} (−ξ_{k}(ν) + ∑_{j=1}^{p} b_{j,k}β_{j}(ν)) = 0,

where k = 1 , …, s; (II) for each κ ∈ ℕ, the system, composed by the infinite sequences

ξ_{k}(κ + 1) , …, ξ_{k}(κ + μ) , …,

where k = 1 , …, s, is linearly independent system over ℝ.

Proof. Let h_{1} , …, h_{p} is a standard basis of ℝ^{p} (i.e. h_{1} , …, h_{p} are columns of a unit p × p − matrix ). Let further

(2.2.48)  g_{k} = ∑_{j=1}^{p} b_{j,k}h_{j}.

Then

(2.2.49)  g_{1} , …, g_{s}

are columns of the matrix B. Let 𝔏^{∼} is the linear envelope over the field ℝ of system (2.1.47). We consider ℝ^{p} in the role of the space 𝔏 of the Theorem 2.2.2.

Sufficiency. There exists a unique linear over ℝ function ϕ_{ν} on 𝔏 with the following property:

(2.2.50)  ϕ_{ν}(h_{j}) = β_{j}(ν),

where j = 1 , …, p. According to (2.1.48) and (2.2.50),

(2.2.51)  ϕ_{ν}(g_{k}) = ∑_{j=1}^{p} b_{j,k}ϕ_{(}ν)(h_{j}) = ∑_{j=1}^{p} b_{j,k}β_{j}(ν)

There exists a unique linear over ℝ function ψ_{ν} on 𝔏_{0} with the following property:

(2.2.52)  ψ_{ν}(e_{k}) = ξ_{k}(ν),

where k = 1 , …, s. Since ξ_{k}(ν) ∈ ℤ for k = 1 , …, s and ν ∈ ℕ, it follows that ϕ_{ν} ∈ L_{0}^{∧}. The condition (I) implies the equality (2.2.43), the condition (II) implies the equality (2.2.44).

Therefore (2.2.45) holds.

Necessity. According to assertion of the Theorem 2.2.2, there exists a linear over ℝ map α_{0} of the space 𝔏_{0} into the space 𝔏, there exists a sequence of a linear functions ϕ_{ν} ∈ 𝔏′ (where ν ∈ ℕ ) and there exists a sequence of the linear functions ψ_{ν} ∈ L_{0}^{∧} (where ν ∈ ℕ) such that the equalities (2.2.43) and (2.2.44) hold. Let

(2.2.53)  β_{j}(ν) = ϕ_{ν}(h_{j}),

where j = 1 , …, p, and ν ∈ ℕ,

(2.2.54)  ξ_{k}(ν) = ψ_{ν}(e_{k}),

where k = 1 , …, s and ν ∈ ℕ. Since ψ_{ν} ∈ L_{0}^{∧}, it follows that

(2.2.55)  ξ_{k}(ν) ∈ ℤ,

where k = 1 , …, s and ν ∈ ℕ. Then

(2.2.56)  ϕ_{ν}(g_{k}) = ϕ_{ν}(g_{k}) = ϕ_{ν}(∑_{i=1}^{p} b_{j,k}h_{i})

∑_{i=1}^{p} b_{i,k}ϕ_{ν}(h_{i}) = ∑_{j=1}^{p} b_{i,k}β_{i}(ν).

The assertion (I) follows from (2.2.43) for f = e_{k} with k = 1 , …, s, (2.2.53), (2.2.54), (2.2.55) and(2.2.56). The equality (2.2.44) is equivalent to the assertion (II). ■

Corollary 2. Let C = (c_{j,k}), j = 1 , …, r, k = 1 , …, s is a r × s − matrix, all the elements of which are complex numbers. In order that the columns of matrix B are linear independent over the field ℚ it is necessary and sufficient the existence of r sequences of complex numbers γ_{1}(ν) , …, γ_{r}(ν) and existence of s sequences of integer numbers ξ_{1}(ν) , …, ξ_{s}(ν) such that the following conditions are fulfilled:

(A)

lim_{ν→∞}(−ξ_{k}(ν) + ∑_{j=1}^{r} Re (c_{j,k}\overline{γ_{j}(ν)}) = 0,

where k = 1 , …, s; (B) for each κ ∈ ℕ, the system, composed by the infinite sequences

ξ_{k}(κ + 1) , …, ξ_{k}(κ + μ) , …,

where μ runs through ℕ and k = 1 , …, s, is linearly independent system over ℝ.

Proof. Let C = U +iV, where U = (u_{j,k}) is a real r×s − matrix and V = (v_{j,k}) is a real r × s − matrix. Let

C^{∗} = ( \begin{matrix}U \\ V \end{matrix} ) .

Clearly, the columns of the matrix C compose a linearly independent system over the field ℚ if and only if the columns of the matrix C^{∗} compose a linearly independent system over the field ℚ. According to Corollary 1, the columns of the matrix C^{∗} compose a linear independent system over the field ℚ if and only if there exist a sequences of real numbers

α_{1}(ν) , …, α_{r}(ν),
β_{1}(ν) , …, β_{r}(ν)

and there exist sequences of integer numbers

ξ_{1}(ν) , …, ξ_{s}(ν)

( where ν ∈ ℕ ) such that the condition (II) of the corollary 1 is fulfilled and, for k = 1 , …, s, the following equality holds:

lim_{ν→∞}(−ξ_{k}(ν) + ∑_{j=1}^{r}(α_{j}(ν)u_{j,k} + β_{j}(ν)v_{j,k})) = 0.

Let γ_{j,k} = α_{j,k} + iβ_{j,k} for j = 1 , …, r, k = 1 , …, s. Then

Re (c_{j,k}\overline{γ_{j}(ν)} = α_{j}(ν)u_{j,k} + β_{j}(ν)v_{j,k}

where j = 1 , …, r, k = 1 , …, s. ■

§3. Non-discrete free modules over the ring ℤ_{K}.

Let, as in paragraph 1, 𝔐_{0} is a free 𝔎 − module with s free generators (2.1.15) and M_{0} is the free ℤ_{K}-module (2.1.16).

Let further M is the free ℤ_{K}-submodule of normed 𝔎-module 𝔐 (which, of course, we may regard as ℤ_{K}-module ).

Let β_{0} is the ℤ_{K}-isomorphism of M_{0} onto M. If for any C > 0 and for any α > 0 there exists an element f ∈ M_{0}\{0} satisfying to the inequality

q(β_{0}(f)) < C(q_{∞}^{(𝔎)}(f))^{−α},

then we name the module M by Liouvillie’s module. In contrary case there are C > 0 and α > 0 such that

q(β_{0}(f)) ≥ C(q_{∞}^{(𝔎)}(f))^{−α}

for any f ∈ M_{0}\{0}; we name the lower bound of such numbers α by the measure of nondiscreteness of module M relative to the norm q and we denote this lower bound by μ_{q}(M).

These definitions are independent from the choice of the isomorphism β_{0}. Let β_{1} another such isomorphism. Let M_{0}^{∼} and M^{∼} are the linear over K spaces generated by M_{0} and M respectively. Clearly,isomorphisms β_{0} and β_{1} can be continued uniquely to the isomorphisms respectively β_{0}^{∼} and β_{1}^{∼} of linear over K space M_{0}^{∼} onto linear over K space M^{∼}. The maps β_{0}^{∼}(β_{1}^{∼})^{−1} and β_{1}^{∼}(β_{0}^{∼})^{−1} (respectively (β_{0}^{∼})^{−1}β_{1}^{∼} and (β_{1}^{∼})^{−1}β_{0}^{∼}) are automorphisms of a linear over K spaces M_{0}^{∼} (respectively M^{∼}), which are finite-dimensional over the field ℚ; of course, the specified automorphisms are automorphisms also over the field ℚ. Consequently, there exists a constant C > 0 satisfying to the inequalities

q_{∞}^{𝔎}((β_{j}^{∼}(β_{1−j}^{∼})^{−1})(f) ≤ Cq_{∞}^{𝔎}(f),
q(((β_{1−j}^{∼})^{−1}β_{j}^{∼})(g)) ≤ Cq(g)

for any g ∈ M^{∼}, for any f ∈ M_{0}^{∼} and for any j = 0, 1. Then

q(β_{j}^{∼}(f)) = q((β_{j}^{∼}((β_{1−j}^{∼})^{−1})β_{1−j}^{∼}(f)) ≤
Cq(β_{1−j}^{∼}(f))

for any f ∈ M_{0}^{∼} and for any j = 0, 1.

If K = ℚ, α is not Liouvillie’s irrational number and M = ℤ + ℤα, then for standard norm q on ℝ the number μ_{q}(M) coincides with the measure of irrationality of the number α. Also, clearly, α is Liouvillie’s number if and only if the ℤ-module M = ℤ + ℤα, is Liouvillie’s module.

I prove the following Lemma with aim to emphasize dependence of the measure of nondiscreteness from the norm q and to show that the name of this notion is natural.

Lemma 2.3.1. Let M is a normed 𝔎-module 𝔐, having finite set of generators and let q is its norm. For the discreteness of ℤ_{K}-submodule M of the module 𝔐 it is necessary and sufficient the realizability of the following condition:

the existence in M of some free ℤ_{K}-submodule M^{∨}, which has a finite index in M and μ_{q}(M^{∨}) = 0.

Proof.

Necessity

Clearly, M contains some basis

f_{1}, , …, f_{s}

of a linear over K envelope M^{∼} of the module M. Let

M^{∨} = ℤ_{K}g_{1} + … + ℤ_{K}g_{s}.

Clearly,M^{∨} has a finite index in M. Clearly, there exists the unique the ℤ_{K}- isomorphism β_{0} of module M_{0} in (2.1.16) onto M^{∨} with the following property:

β_{0}(e_{k}) = f_{k} for k = 1 , …, s.

Since M is discrete relative to the norm q, it follows that M^{∨} is discrete relative to the norm q. Since the module M^{∨} is discrete relative to the norm q, it follows that there exists the number δ > 0 with the following property:

q(g) = q(g − 0) ≥ δ for any g ∈ M^{∨}\{0}.

Then

q(β_{0}(f)) ≥ δ = δ(q_{∞}^{(𝔎)}(f))^{−0}

for any f ∈ M_{0}^{∨} \{0}.

Sufficiency.

Clearly, for the discreteness of module M it is necessary and sufficient the realizability of the following condition:

the discreteness of some submodule M^{∨} of module M and the finiteness of index of this submodule M^{∨} in M.

Therefore we may regard that μ_{q}(M) = 0.

We consider first the case, when the generated by M the 𝔎-submodule 𝔐_{1} of 𝔎-module 𝔐 is free 𝔎 − module. In this case, according to the Lemma 2.4.4, the isomorphism β_{0} can be chosen such that

(2.3.1)  g_{1} = β_{0}(e_{1}) , …, g_{s} = β_{0}(e_{s})

and m of last elements of system (2.3.1) are free generators of 𝔐_{1}. Let A is the m × s-matrix composed by columns of coordinates of elements of system (2.3.1) in the basis

(2.3.2)  g_{n+1} , …, g_{s}.

If m < s, then, according to the Lemma 2.2.2, there are C_{1} > 0 and for each ν ∈ ℕ a column

X_{ν} = ( x_{1}(ν) ; ⋮ ; x_{s}(ν) ) ∈ (ℤ_{K})^{s}\{0}

such that q_{∞}^{(𝔎)}(X) ≤ C_{1}(ν)^{m/(s−m)} and for

Y_{ν} = ( y_{1}(ν) ; ⋮ ; y_{m}(ν) ) = AX

the following inequality holds:

(2.3.3)  q_{∞}^{(𝔎)}(Y_{ν}) ≤ C_{1}/nu.

Therefore

q_{∞}^{(𝔎)}(Y_{ν}) ≤ (C_{1})^{s/m}(q_{∞}^{(𝔎)}(X_{ν}))^{−(s−m)/m}.

Let

f_{ν} = ∑_{k=1}^{s} x_{k}(ν)e_{k},

h_{ν} = ∑_{i=1}^{m} y_{i}(ν)g_{n+i}.

Then f_{ν} ≠ 0, h_{ν} = β_{0}(f_{ν}) ≠ 0 and

0 < q(β_{0}(f_{ν})) = q(h_{ν}) = q (∑_{i=1}^{m} y_{i}(ν)g_{n+i}) ≤ ∑_{i=1}^{m} q(y_{i}(ν)g_{n+i}) ≤

∑_{i=1}^{m} q_{∞}^{(𝔎)}(y_{i}(ν))q(g_{n+i}) ≤

(C_{1})^{s/m}(q_{∞}^{(𝔎)}(f_{ν}))^{−(s−m)/m} ∑_{i=1}^{m} q(g_{n+i}).

Let ε ∈ (0, (s − m)/m). Since μ_{q}(M) = 0, it follows that there exists C_{2} > 0 with the following property:

C_{2}(q_{∞}^{(𝔎)}(f_{ν}))^{−ε} ≤ q(β_{0}(f_{ν})).

Then

(2.3.4)  (q_{∞}^{(𝔎)}(f_{ν}))^{((s−m)/m)−ε} ≤ ((C_{1})^{s/m}/C_{2}) ∑_{i=1}^{m} q(g_{n+i}).

But this is incompatible with (2.3.3) because M_{0} is discrete ℤ_{K}-module and the set of all its elements satisfying (2.3.4) is finite. Therefore the case s > m is impossible. Let s = m. Then 𝔐_{1} is a free 𝔎-module with the s free generators (2.3.2), where n = 0. There exists a unique 𝔎-homomorphism β_{0}^{∗} of module 𝔐_{0} into 𝔐_{1} with the following property:

β_{0}^{∗}(e_{k}) = g_{k} for k = 1 , …, s.

Since (2.3.2) with n = 0 is free system of 𝔎-generators of 𝔐_{1}, it follows that homomorphism β_{0}^{∗} is isomorphism, which on M coincides with β_{0} and on M^{∼} - with β_{0}^{∼}. Let q^{∼} is the function defined on the module 𝔐_{1} by means of the equality:

q^{∼}(g) = q_{∞}^{(𝔎)}((β_{0}^{∗})^{−1}(g))

for any g ∈ 𝔐_{1}. Then q^{∼} is the norm on 𝔐_{1} and, clearly, M is discrete relative to q^{∼}. But the norms q and q^{∼} are equivalent because they are defined by means their restrictions on finite-dimensional real or complex linear spaces, which are direct summands of the module 𝔐_{1}.

Therefore M is discrete also relative to the norm q.

We consider, finally, the general case. The discreteness (respectively nondiscreteness) of the set M, as a topological notion, is independent from algebraical structure on this set. Therefore we can consider M_{0} (respectively M) not only as a free ℤ_{K}-submodule of 𝔎-module 𝔐_{0} (respectively 𝔐) with the free ℤ_{K}-generators (2.1.15) (respectively (2.3.1)) but also as a free ℤ-submodule of ℝ-module 𝔐_{0} ( respectively 𝔐) with free ℤ − generators (2.1.17)

(respectively

(2.3.5)  ω_{k}g_{l},

where k = 1 , …, d, l = 1 , …, d. ) The system (2.1.17) is also the system of free ℝ-generators of free ℝ-module 𝔐_{0} (i.e. basis of linear space 𝔐_{0} over the field ℝ). The ℤ_{K}-isomorphism β_{0} maps the basis (2.1.17) onto the basis (2.3.5). Since the norms q_{∞}^{(𝔎)} and q_{∞}^{(𝔅)} are equivalent, as the norms of free 𝔎-module 𝔐_{0} with finite set of a free 𝔎-generators, the equality

μ_{q}^{(𝔎)}(M) = μ_{q}^{(ℝ)}(M)

follows from the equality (2.1.9) and, if μ_{q}^{(𝔎)}(M) = 0, it follows that

μ_{q}^{(ℝ)}(M) = 0.

Since generated by means of the basis (2.3.5) ℝ-module, as linear space over the field ℝ, is free and coincides with 𝔐_{1}, it follows that, according to the first part of proof, the set M is discrete set in 𝔐 relative to the norm q. ■

For each Z ∈ 𝔎, let

∥Z∥_{K} = inf_{W ∈ℤ_{K}}{q_{∞}^{(𝔎)}(Z − W)}

Lemma 2.3.2 If Z ∈ 𝔎 and q_{∞}^{(𝔎)}(Z) ≤ \frac{1}{2}, then

∥Z∥_{K} = q_{∞}^{(𝔎)}(Z).

Proof. If, for any W ∈ ℤ_{K}, the inequality

q_{∞}^{(𝔎)}(Z − W) < q_{∞}^{(𝔎)}(Z)

holds, then

q_{∞}^{(𝔎)}(W) ≤ q_{∞}^{(𝔎)}(Z) + q_{∞}^{(𝔎)}(Z − W) < 2q_{∞}^{(𝔎)}(Z) ≤ 1

and, according to (2.1.4), W = 0. blacksqare Let m ∈ ℕ, n ∈ ℕ,

(2.3.6)  a_{i,k} ∈ 𝔎

for i = 1 , …, m, k = 1 , …, n,

(2.3.7)  α_{l}^{∧}(ν) ∈ ℤ_{K},

where l = 1 , …, m + n and ν ∈ ℕ. Let further there are γ_{0}, r_{1}^{∧} ≥ 1 , …, r_{m}^{∧} ≥ 1 such that

(2.3.8)  q_{∞}^{(𝔎)}(α_{i}(ν)) < γ_{0}(r_{i}^{∧})^{ν},

where i = 1 , …, m and ν ∈ ℕ. Let

(2.3.9)  y_{k}(ν) = −α_{m+k}^{∧}(ν) + ∑_{i=1}^{m} a_{i,k}α_{i}^{∧}(ν),

where k = 1 , …, n and ν ∈ ℕ. If X = ( Z_{1} ; ⋮ ; Z_{n} ) ∈ (𝔎)^{n}, then let

(2.3.10)  y^{∧}(X) = y^{∧}(X, ν) = ∑_{k=1}^{n} y_{k}^{∧}(ν)Z_{k}

for ν ∈ ℕ, let

(2.3.11)  ϕ_{i}(X) = ∑_{k=1}^{n} a_{i,k}Z_{k}

for i = 1 , …, m and ν ∈ ℕ and let

(2.3.12)  α_{0}^{∧}(X, ν) = ∑_{k=1}^{n} α_{m+k}^{∧}(ν)Z_{k}

for ν ∈ ℕ. Clearly,

(2.3.13)  y^{∧}(X, ν) = −α_{0}^{∧}(X, ν) + ∑_{i=1}^{m} α_{i}^{∧}(ν)ϕ_{i}(X)

for X ∈ (𝔎)^{n} and ν ∈ ℕ,

(2.3.14)  α_{0}^{∧}(X, ν) =∈ ℤ_{K}

for X ∈ (ℤ_{K})^{n} and ν ∈ ℕ.

Now I can proceed to main result of this part:

Theorem 2.3.1.Let l ∈ ℕ, γ_{1} > 0, let

(2.3.15)  γ_{2} > \frac{1}{2}, R_{1} ≥ R_{2} > 1,

let

(2.3.16)  α_{i} = (log(r_{i}^{∧}R_{1}))/ log(R_{2}),

where i = 1 , …, m, let

(2.3.17)  γ_{3} = γ_{1}(R_{1})^{(− log(2γ_{2}R_{2}))/ log(R_{2})},

(2.3.18)  γ_{4} = γ_{3} (∑_{i=1}^{m} γ_{0}(r_{i}^{∧})^{(log(2γ_{2}))/ log(R_{2})+l})^{−1} ,

let

(2.3.19)  X ∈ (ℤ_{K})^{n}\{(0)},

and let for each ν ∈ ℕ − 1 the following inequalities hold:

(2.3.20)  γ_{1}(R_{1})^{−ν}q_{∞}^{(𝔎)}(X) ≤ sup{q_{∞}^{(𝔎)}(y^{∧}(X, κ)): κ = ν , …, ν + l − 1))},

(2.3.21)  q_{∞}^{(𝔎)}(y^{∧}(X, ν)) ≤ γ_{1}(R_{2})^{−ν}q_{∞}^{(𝔎)}(X).

Then

(2.3.22)  sup{∥ϕ_{i}(X)∥_{K}(q_{∞}^{(𝔎)}(X))^{α_{i}}: i = 1 , …, m} ≥ γ_{4}.

Proof. According to (2.3.15), (2.3.19) and (2.1.4),

(2.3.23)  2γ_{2}q_{∞}^{(𝔎)}(X) > 1

32 LEONID A. GUTNIK

Let

(2.3.24)  ν_{1} = inf{ν ∈ ℕ − 1: γ_{2}q_{∞}^{(𝔎)}(X)(R_{2})^{−ν} ≤ 1/2} =
inf{ν ∈ ℕ − 1: ν ≥ log(2γ_{2}q_{∞}^{(𝔎)}(X))/ log(R_{2})

Clearly,

ν_{1} = log(2γ_{2}q_{∞}^{(𝔎)}(X))/ log(R_{2}) + θ,

where θ ∈ [0, 1),

(2.3.25)  γ_{1}(R_{1})^{−ν_{1}}q_{∞}^{(𝔎)}(X) =

γ_{1} exp(−(log(R_{1}))(log(2γ_{2}q_{∞}^{(𝔎)}(X))/ log(R_{2}) + θ))q_{∞}^{(𝔎)}(X) =

γ_{1}(q_{∞}^{(𝔎)}(X))^{−(log((R_{1})/R_{2}))/ log(R_{2})}×

exp(−(log(R_{1}))((log(2γ_{2}))/ log(R_{2}) + θ)) ≥ γ_{1}(exp(−(log(R_{1}))(log(2γ_{2}R_{2}))/ log(R_{2})))×

(q_{∞}^{(𝔎)}(X))^{−(log((R_{1})/R_{2}))/ log(R_{2})} =
γ_{3}(q_{∞}^{(𝔎)}(X))^{−(log((R_{1})/R_{2}))/ log(R_{2})}.

In view of (2.3.24),

(2.3.26)  γ_{2}q_{∞}^{(𝔎)}(X)(R_{2})^{−ν} ≤ 1/2

for ν ∈ ν_{1} + ℕ − 1. Let ν_{2} ∈ {ν_{1} , …, ν_{1} + l − 1} and

q_{∞}^{(𝔎)}(y^{∧}(X, ν_{2})) = sup{q_{∞}^{(𝔎)}(y^{∧}(X, κ)): κ = ν_{1} , …, ν + l − 1}.

Then, in view of (2.3.20), (2.3.25), (2.3.19), (2.3.24) and (2.3.8),

\frac{1}{2} ≥ q_{∞}^{(𝔎)}(y^{∧}(X, ν_{2})) ≥ γ_{1}(R_{1})^{−ν_{1}}q_{∞}^{(𝔎)}(X) ≥

γ_{3}(q_{∞}^{(𝔎)}(X))^{−(log((R_{1})/(R_{2})/ log(R_{2})}.

Let f_{1} ∈ ℤ_{K} , …, f_{m} ∈ ℤ_{K} are chosen such that

(2.3.27)  q_{∞}^{(𝔎)}(ϕ_{i}(X) − f_{i}) = ∥ϕ_{i}(X)∥_{K},

and let

(2.3.28)  g = −α_{0}^{∧}(X, ν_{2}) + ∑_{i=1}^{m} α_{i}^{∧}(ν_{2})f_{i}.

ON THE DIMENSION OF SOME LINEAR SPACES OVER FINITE EXTENSION OF ℚ 33

Then, in view of(2.3.12), (2.3.27), and, according to the Lemma 2.3.2, g ∈ ℤ_{K} and

γ_{3}(q_{∞}^{(𝔎)}(X))^{−(log((R_{1})/R_{2}))/ log(R_{2})} ≤ q_{∞}^{(𝔎)}(y^{∧}(X, ν_{2}) =

∥q_{∞}^{(𝔎)}(y^{∧}(X, ν_{2}))∥_{K} ≤ q_{∞}^{(𝔎)}(y^{∧}(X, ν_{2}) − g) ≤

∑_{i=1}^{m} q_{∞}^{(𝔎)}(α_{i}^{∧}(ν_{2}))q_{∞}^{(𝔎)}(ϕ_{i}(X) − f_{i}) =

∑_{i=1}^{m} q_{∞}^{(𝔎)}(α_{i}^{∧}(ν_{2}))∥ϕ_{i}(X)∥_{K} ≤

∑_{i=1}^{m} ∥ϕ_{i}(X)∥_{K}γ_{0}(r_{i}^{∧})^{ν_{1}+l−1}.

But

(r_{i}^{∧})^{ν_{1}} = exp((log(r_{i}^{∧}))((log(2γ_{2}q_{∞}^{(𝔎)}(X)))/ log(R_{2}) + θ)) =
(q_{∞}^{(𝔎)}(X))^{(log(r_{i}^{∧}))/ log(R_{2})}(r_{i}^{∧})^{(log(2γ_{2}))/ log(R_{2}))+θ))},

where i = 1 , …, m. Therefore γ_{3}(q_{∞}^{(𝔎)}(X))^{−(log((R_{1})/(R_{2})/ log(R_{2})} ≤ q_{∞}^{(𝔎)}(y^{∧}k(X, ν_{2}) ≤

sup{∥ϕ_{i}(X)∥_{K}(q_{∞}^{(𝔎)}(X))^{(log(r_{i}^{∧}))/ log(R_{2})}: i = 1 , …, m}×
(2.3.29)  ∑_{i=1}^{m} γ_{0}(r_{i}^{∧})^{(log(2γ_{2}))/ log(R_{2})+l}

The inequality (2.3.22) follows from (2.3.29) . ■ This Theorem was obtained in [28] (Lemma 25) and, if the number field coincides with ℚ, in [26] (Theorem 8).

§4.Survey of some results.

It is known that for Riemann’s ζ− function

ζ(s) = ∑_{n=1}^{∞} n^{−s},

the following equality holds:

ζ(2k) = B_{2k}(2π)^{2k}(−1)^{k+1}/(2(2k)!), where k ∈ ℕ an B_{2k} ∈ ℚ(k ∈ ℕ) are the Bernoulli’s Numbers . But about the numbers ζ(2k + 1)(k ∈ ℕ) it was known nothing till 1978, when R.Apéry [1] has proved that the number ζ(3) is irrational.

In 1979, [21], [22] author by means of Meijer’s functions proved the following Theorem:

Theorem 2.4.1. Let M is the ℤ− module with generators

(2.4.1.)  ( \begin{matrix}log(4) \\ ζ(2) \end{matrix} ) , ( \begin{matrix}ζ(2) \\ 3ζ(3) \end{matrix} ) , ( \begin{matrix}1 \\ 0 \end{matrix} ) , ( \begin{matrix}0 \\ 1 \end{matrix} )

Then the columns (2.4.1) are ℤ− free generators of module M, the module M is not Liouvillie’s module and has the measure of nondiscreteness not bigger than 106, 1019.

Later [27] the estimation specified in the Theorem 2.4.1 was decreased up to 22, 4271.

Let

a_{1} = d_{1} = log(2), c_{1} = −b_{1} = −\frac{π}{2} , γ^{(1)} = 3, 3887,

a_{2} = d_{2} = log(4/3), c_{2} = −3b_{2} = −\frac{π}{\sqrt{3}}, γ^{(2)} = 2, 6161,

a_{3} = d_{3} = log(3), c_{3} = −3b_{3} = −\frac{π}{2} , γ^{(3)} = 3, 0736,

Let further

(2.4.2)  f_{i,1} = ( \begin{matrix}a_{i} \\ c_{i} \end{matrix} ) , f_{i,2} = ( \begin{matrix}b_{i} \\ d_{i} \end{matrix} ) , f_{i,3} = ( \begin{matrix}1 \\ 0 \end{matrix} ) , f_{i,4} = ( \begin{matrix}0 \\ 1 \end{matrix} )

where i = 1, 2, 3, and M_{i}, is the ℤ− module with four generators (2.4.2). From Baker’s Theorem follows that the generators (2.4.2) are a free ℤ-generators of ℤmodule M_{i}. The following Theorem was proved in [28] :

Theorem 2.4.2. For i = 1, 2, 3 the measure of nondiscreteness of M_{i} is not bigger than γ^{(i)}.

The following Theorem, which is proved in [29], may be mentioned in connection with Theorem 2.3.1:

Theorem 2.4.3. Let m = 2, 3 , …, 37, let K = ℚ[exp(2πi/m] let Gal (K/ℚ) is the Galois group of the extension K/ℚ, let log(z), where z ∈ ℂ\(∞, 0] is the branch of Log (z) with | arg(z)| < π and let α_{m} is the (m − 1)−th element of the following sequence of real numbers (the numbers in this sequence are separated each from other by means of semicolon):

2, 8913997710; 3, 0735255571; 3, 3886633478; 4, 7846293625;

4, 0874625966; 6, 7393917326; 7, 6354250465; 9, 0606062958;
9, 5089288012; 11, 890912329; 13, 408522861; 15, 302385909;
16, 312767157; 18, 878054599; 20, 789878818; 23, 159619505;
25, 015235814; 28, 383873057; 31, 232455499; 34, 903849079;
38, 193794800; 43, 271329176; 48, 284195791; 54, 400694383;
60, 675050525; 69, 926378752; 80, 085043105; 93, 086523748;
108, 61532969; 131, 33065226; 161, 36496124; 206, 46682440;
278, 01614475; 421, 43377055; 808, 33395542; 6355, 0702897.

Then there exists C > 0 with the following property:

max_{σ∈Gal (Kℚ)}(|q^{σ} log(1 − 1/z^{σ}) − p^{σ}|) ≥

(2.4.2)  C( max_{σ∈Gal (Kℚ)}(|q^{σ}|))^{−α_{m}}

The estimates obtained by Hata [16] for |qlog2 − p| and |q \frac{π}{\sqrt{3}} − p| by means of another method follow from (2.4.2) for m = 2 and m = 6. The cases m = 3 and m = 4 are considered in [28]. These problems are a part of a more general problem of Diophantine Approximations of polylogarithms, i.e. of functions of a kind

L_{s}(z) = ∑_{n=1}^{∞} n^{−s}z^{n}.

Nikishin [5] has proved the following Theorem ( I follow in the formulation the paper [15] ) :

Nikishin’s Theorem. Let (a, b) ∈ ℤ × ℕ, −a > (4es)^{s(s−1)}b^{s+1)} for some s ∈ ℕ. Then the numbers

1, L_{1}(b/a), L_{2}(b/a), …, L_{s}(b/a)

compose a linearly independent system over the field ℚ.

This result was improved and generalized in [23], [24], [25].

Let m ∈ ℕ, n ∈ ℕ, a ∈ ℤ\{0}, b ∈ ℕ and let Λ_{m,n}(b/a) = (a_{i,j}(b/a)) denotes the m × n− matrix, whose element at the intersection of its i-th row and its j-th column is equal to

a_{i,j}(b/a) = \binom{i + j − 2}{i − 1}L_{i+j−1}(b/a),

for i = 1 , …, m and j = 1 , …, n. Let further E_{m} is the unit m × m− matrix. I have proved in [24], [25] that, if

|a| > (\frac{m}{n})^{m} exp (\frac{n}{m} (n + (m − 1) max (\frac{m}{n}, 1))) b^{\frac{m+n}{m}} ,

then the ℤ− module M generated by columns of the matrix

(Λ_{m,n}(b/a), E_{m})

is ℤ− module with m + n free ℤ− generators. I have proved in [26] that it is not a Liouville module and have found a upper estimate of the measure of nondiscreteness of this ℤ-module.

I hope to give, in the next parts, the proofs of certain results mentioned above and also of some other results .

References

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[3] van der Porten A., A proof that Euler missed...Apéry’s proof of the irrationality of ζ(3), 1 (1979), Math Intellegencer, 195 – 203..

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[10] , Un systḿe explicite d’approximants de Padé pour les fonctions hypérgéometriques généraliesées,avec applications a l’arithmétique,, C.R.Acad.Sc. Paris,Série A t.288 (1979), 1001 – 1004.

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[15] S.Eckmann, Über die lineare Unaqbhangigkeit der Werte gewisser Reihen, Results in Mathematics, 11 (1987), 7 – 43.

[16] Hata M.,1990., Legendre type polinomials and irrationality mesures,, J. Reine Angew. Math. 407 pages 99 – 105 (1990).

[17] A.O. Gelfond., Transcendental and algebraic numbers. (Russian), GIT-TL, Moscow, 1952.

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[21] L.A.Gutnik L., On the Irrationality of Some Quantyties Containing ζ(3) (Russian),, Uspekhi Mat. Nauk 34, 3(207) (1979), 190.

[22] , Eleven papers translated from the Russian, vol. 140, American Mathematical Society, 1988, pp. 45 - 56.

[23] , Linear independence over Q of dilogarithms at rational points (Russian), Uspekhi Mat.Nauk 37 (1982), 179-180; English transl. in Russ. Math. surveys 37 (1982), 176-177..

[24] , On the rank over ℚ of some real matrices, 5736-84 (1984), VINITI, 1 – 29.

[25] , On the rank over ℚ of some real matrices, 27 (2002), Max-Plank-Institut für Mathematik, Bonn, Preprint Series, 1 – 32.

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[27] , To the question of the smallness of some linear forms, 2413-B93 (1993), VINITI, 1 – 94.

[28] ,1995., About linear forms,whose coefficients are logarithms of algebraic numbers, No 135-B95 (1995), VINITI, 1 – 149.

[29] , About systems of vectors, whose coordinates are linear combinations of logarithms of algebraic numbers with algebraic coefficients. 3122-B94 (1994), VINITI, 1 – 158.

[30] , About linear forms, whose coefficients are ℚ-proportional to the number log 2, and the values of ζ(s) for integer s, 3258-B96 (1996), VINITI, 1 – 70.

[31] , The lower estimate for some linear forms with coefficients ℕζ(1 + ℕ), 3072-B97 (1997), VINITI, 1 – 77.

[32] , About the behaviour of solutions of difference equations of Poincaré type, 3384-B97 (1997), VINITI, 1 – 41.

[33] , On some linear forms with coefficients in ℕζ(1 + ℕ), 3 (2000), Max-Plank-Institut für Mathematik, Bonn, Preprint Series, 1 – 13.

[34] , On some linear forms with coefficients in ℕζ(1 + ℕ), (the detailed version,part 1) 15 (2001), Max-Plank-Institut für Mathematik, Bonn, Preprint Series, 1 – 20.

[35] , On some linear forms with coefficients in ℕζ(1 + ℕ) (the detailed version,part 2) 104 (2001), Max-Plank-Institut für Mathematik, Bonn, Preprint Series, 1 – 36.

[36] , On some linear forms with coefficients in ℕζ(1 + ℕ) (the detailed version,part 3) 57 (2002), Max-Plank-Institut für Mathematik, Bonn, Preprint Series, 1 – 33.

Moscow State Institute of Electronics and Mathematics Bolshoi Trekhsvyatitelskij pereulok, 1-3/12, str.8 Moscow 109028 Russian Federation E-mail: gutnik@gutnik.mccme.ru