國立交通大學
應用數學系
碩 士 論 文
正規 Laurent 級數體上探討 Kurzweil 定理
Kurzweil’s Theorem in the Field of Formal Laurent Series
研 究 生: 陳書誼
正規 Laurent 級數體上探討 Kurzweil 定理
Kurzweil’s Theorem in the Field of Formal Laurent Series
研 究 生: 陳書誼 Student: Shu-Yi Chen
指導教授: 符麥克 Advisor: Michael Fuchs
國 立 交 通 大 學
應用數學系
碩 士 論 文
A Thesis
Submitted to Department of Applied Mathematics National Chiao Tung University
in Partial Fulfillment of the Requirements for the Degree of
Master in
Applied Mathematics
June 2012
Hsinchu, Taiwan, Republic of China
Kurzweil’s Theorem in the field of
Formal Laurent series
Shu-Yi Chen
Department of Applied Mathematics,
National Chiao Tung University
前言
我們將在本論文中探討正規 Laurent 級數體之下的賦距同步 Diophantine 逼近。在最近的一篇論文中，Kim 和 Nakada 證明了在一維度的正規 Laurent 級數之下，和 Kurzweil 定理相似的一個結果。本論文主要的工作是提供一個新 的證明方法，甚至可推廣到同步 Diophantine 逼近。 本文的主要架構如下：我們將在第一章介紹 Diophantine 逼近的背景。此 章分為三小節。第一節，我們簡單地回顧 Diophantine 逼近和賦距 Diophantine 逼近的概念，並說明一些在實數體下的結論及在正規 Laurent 級數體下的相似 結果。在第二節中，我們將介紹非齊次同步 Diophantine 逼近的概念。此外， 我們羅列了一些定義和符號，以及關於所謂的 double-metric 和 single-metric 的 結論。最後，第三節包含了我們本論文主要的結果。 在第二章，我們回顧一些在正規 Laurent 級數體之下的基本性質。 而第三、四章包含了 0-1 法則和一連串對於我們在一維、高維度的證明 過程中非常重要的引理。而我們主要結果的證明就是根據這些引理得證。事實 上，在一維度的結果即是高維度結論中的一個特例，但為了方便閱讀以及為了 高維度證明的想法做準備，我們將優先處理一維度的例子。 最後，在第五章，我們將針對本論文做一個總結。
Preface
This thesis is concerned with metric simultaneous Diophantine approxima-tion in the field of formal Laurent series. In a recent paper, Kim and Nakada proved an analogue of Kurzweil’s theorem in dimension one for formal Laurent series. The main aim of this thesis is to give a new proof which works for simul-taneous Diophantine approximation as well.
An outline of this thesis is as follows. In Chapter 1, we will introduce background on Diophantine approximation. This chapter is split into three sec-tions. In Section 1.1, we will briefly recall Diophantine and metric Diophantine approximation, and state some results in the real case and some analogues over the field of formal Laurent series. Then, in Section 1.2, we will introduce inho-mogeneous (simultaneous) Diophantine approximation. Moreover, we will collect notations and results for the so-called double-metric and single-metric cases. Fi-nally, Section 1.3 will contain our main results. In Chapter 2, we will recall some fundamental properties for formal Laurent series. Chapter 3 and Chapter 4 will contain zero-one laws and a series of lemmas which are important for the proof of our results in dimension one and higher dimension, respectively. The proofs will follow from these lemmas. We want to point out that the result in dimension one is in fact only a special case of the higher dimensional result. Nevertheless, for the sake of readability and as a warm-up, we will treat the one-dimensional case separately. Finally, we will end the thesis with some concluding remarks in Chapter 5.
致 謝 辭
剛進入交大似乎還記憶猶新，時間卻在不知不覺中飛逝，碩士班生活即 將畫下句點。回顧過去這兩年的時光，首先誠摯的感謝指導老師―符麥克教授， 讓我在碩士班學習過程得以順利，以及細心地指導我論文寫作。感謝兩位口試 委員―楊一帆教授及蕭守仁教授，針對這份論文提出寶貴的建議及通過碩士資 格的審核。 在這兩年的求學道路上，特別感謝系上老師們及行政人員的幫助，以 及建偉、智龍、囿丞等同學在課業上的討論及指導。也感謝女朋友佳穎背後默 默的支持與鼓勵，讓我的碩士班生活過得更加順利。 我知道，這不僅為碩士班的日子畫下句點，也是下一個人生目標的起跑 點。希望可以秉持著在碩士班學到的那份努力，在未來的道路上盡情揮灑。 最後，謹以此文獻給我摯愛的家人。 書誼 謹誌Contents
1 Introduction and Background 1
1.1 (Metric) Diophantine Approximation . . . 1 1.2 Double-metric and Single-metric Inhomogeneous Diophantine
Ap-proximation . . . 4 1.3 Kurzweil’s Theorem in the Field of Formal Laurent Series . . . . 9
2 Preliminaries 11
2.1 Fundamental Properties in Dimension One . . . 11 2.2 Fundamental Properties in Higher Dimension . . . 14
3 Kurzweil’s Theorem in Dimension One 18 4 Kurzweil’s Theorem in Higher Dimension 27 5 Conclusion 37
Chapter 1
Introduction and Background
In this chapter, we will give a historical discussion and discuss recent results related to this research.
1.1
(Metric) Diophantine Approximation
It is well-known that the set of rational numberQ is dense in R. This means that for each x∈ R, there exists a rational sequence {rn} such that |x − rn| < n1,
∀n. A fundamental task is to approximate real numbers by a rational sequence
with good accuracy, where the accuracy is measured in terms of the size of the denominator. The area which is concerned with such investigations is called
Diophantine approximation. In this area, an important question is as follows: for
a fixed irrational number α, which function ψ will make the inequality α − m n < ψ(n) n , m, n∈ Z
have infinitely many solutions m and n? The following is a typical result.
Theorem 1.1 (G. L. Dirichlet). Let α be an irrational number. Then
α − m n < 1 n2, m, n∈ Z (1.1)
has infinitely many solutions m and n.
In this theorem, Dirichlet took ψ(n) = 1
n such that (1.1) has infinitely many
called metric Diophantine approximation, on the other hand, asks for properties which hold for almost all real numbers α. Here, a property holds almost all if the set of elements for which the property does not hold is a null set, that is, a set of Lebesgue measure zero. A famous result in metric Diophantine approximation is the following theorem of Khintchine.
Theorem 1.2 (A. Khintchine). Let ψ(x) be a positive continuous function and
suppose that xψ(x) is non-increasing. Then
α − m n < ψ(n) n , m, n∈ Z
has infinitely many solutions for almost all α∈ R if and only if ∑∞n=1ψ(n) =∞.
There are many other results in the case of real numbers. In this research, we are concerned with analogues in the field of formal Laurent series. Next, we will fix some notation and introduce Diophantine approximation and metric Diophantine approximation for formal Laurent series.
We denote byFq a finite field with q elements, where q = pn, n∈ N, p ∈ P.
Moreover, we denote by Fq[X] the set of polynomials with coefficients inFq, and
by Fq(X) the quotient set of Fq[X]. The elements in this set are called rational.
Finally, we denote by Fq (( X−1))= { f = l ∑ n=−∞ anXn : al ̸= 0, an ∈ Fq } ∪ {0}
the set of formal Laurent series. Next, we consider in Fq((X−1)) an addition and
multiplication, where both operations are defined as for polynomials. Then, the resulting structure is easily seen to be a field. Moreover, we equipFq((X−1)) with
a norm in the standard way, namely, |f| = qdeg(f ) for f ̸= 0 and |0| = 0 (here,
deg(f ) denotes the generalized degree function). In the sequel, the following set will be of importance
L ={f ∈ Fq((X−1)) :|f| < 1
}
.
Restricting the above norm to this set gives a compact topological group. Hence, there exists a unique translation-invariant probability measure which we are going to denote by m.
Similar to the real case, we can study now Diophantine approximation and metric Diophantine approximation in the field of formal Laurent series, where elements of Fq[X] play the role of integers. Fq(X) is dense in Fq((X−1)), i.e., for
each f ∈ Fq((X−1)), there exists a sequence {rn} ⊆ Fq(X) such that
|f − rn| <
1
qn, ∀n.
Again as in the real case, an important task is to approximate the value of f ∈ Fq((X−1)) by{rn} with good accuracy, where the accuracy is measured in terms
of the size of the denominator. This area is called Diophantine approximation
in the field of formal Laurent series. In particular, the analogue to the problem
above is as follows: for fixed f ∈ L, which function ψ will make the Diophantine inequality
f − PQ < ψ(|Q||Q|), P, Q∈ Fq[X] (1.2)
have infinitely many solutions P and Q? The following result is an analogue of Dirichlet theorem for formal Laurent series.
Theorem 1.3 (Analogue of Dirichlet’s Theorem for Formal Laurent Series). We
have,
f − PQ < |Q|12, P, Q ∈ Fq[X]
has infinitely many solutions P and Q.
We will prove Theorem 1.3 in the next chapter. The subarea called metric
Diophantine approximation in the field of formal Laurent series asks for properties
which hold for almost all f ∈ L. In this setting, an analogue of Khintchine’s theorem for formal Laurent series was proved by Fuchs in [1].
Theorem 1.4 (M. Fuchs [1]). Let ψ :{qt: t∈ Z
≥0} → {qt : t∈ Z} be a function
with |Q| ψ(|Q|) non-increasing. Then the inequality (1.2) has infinitely many solutions P and Q for almost all f ∈ L, if and only if
∞
∑
k=0
qkψ(|X|k) =∞.
Moreover, in [4], Inoue and Nakada improved this by dropping the mono-tonicity condition ” |Q| ψ(|Q|) non-increasing” .
Theorem 1.5 (K. Inoue and H. Nakada [4]). Let ψ :{qt: t∈ Z
≥0} → {qt: t∈ Z}
be a function. Then for any set S of positive integers, the inequality (1.2) has infinitely many solutions P and Q for almost all f ∈ L, if and only if
∑
k∈S
qkψ(|X|k) =∞.
In analogy with the integer part of real numbers, we denote by [ g ] the polynomial part of g for all g ∈ Fq((X−1)), i.e., the part of the expansion for
which no negative exponents occur. And we denote by{g} = g−[ g ] the fractional part of g. Note that |{g}| ≤ 1. Then, the inequality (1.2) can be rewritten to
|Qf − P | < ψ(|Q|), P, Q ∈ Fq[X]
which, if ψ(|Q|) ≤ 1, is equivalent to
|{Qf}| < ψ(|Q|), Q ∈ Fq[X].
So far, what we have discussed the so-called homogeneous case. The major in-vestigations in this research will, however, be for the inhomogeneous case. Thus, we will introduce metric inhomogeneous Diophantine approximation in the field of formal Laurent series next.
1.2
Double-metric and Single-metric
Inhomoge-neous Diophantine Approximation
Here, we will introduce the metric inhomogeneous Diophantine approxima-tion. Let us consider the Diophantine inequality
|{Qf} − g| < ψ(|Q|), Q ∈ Fq[X], (1.3)
where f, g ∈ L, and ψ is a {qt: t∈ Z
≥0} → {qt : t∈ Z} function. We will be
concerned with the question of the existence of infinitely many solutions to (1.3) as well as the asymptotic number of solutions as |Q| grows. This area is called
[9], Ma and Su investigated the problem of (1.3) if f and g are both chosen randomly. Let
W (ψ) = {(f, g)∈ L2 : (1.3) has infinitely many solutions Q∈ Fq[X]
}
.
Then, Ma and Su proved the following result.
Theorem 1.6 (C. Ma and W.-Y. Su [9]). Let ψ : {qt : t∈ Z
≥0} → {qt: t∈ Z}
be non-increasing. Then, we have
(m× m) (W (ψ)) = 0, if ∑Q∈F q[X]ψ(|Q|) < ∞, 1, if ∑Q∈F q[X]ψ(|Q|) = ∞.
Moreover, let us consider the inequality whose ψ(|Q|) is equal to q−n−ln
with ln≥ 0 such that
|{Qf} − g| < 1
qn+ln, Q∈ Fq[X], Q monic, n = deg(Q), (1.4)
where f, g ∈ L. In [2], Fuchs investigated the problem of (1.4) and derived strong laws of large numbers with error terms for the number of solutions Q of this inequality with deg(Q) ≤ N. In order to state his result, define
Ψ(N ) := ∑
n≤N
1
qln.
Then, his result reads as follows.
Theorem 1.7 (M. Fuchs [2]). For almost all (f, g)∈ L2, the number of solutions
of (1.4) with 0 ≤ deg(Q) ≤ N satisfies
Ψ(N ) +O ( (Ψ(N ))12 (log Ψ(N )) 3 2+ϵ ) ,
where ϵ > 0 is an arbitrary constant.
These results are for f and g both random. This is the so-called
double-metric case. Moreover, the following two single-double-metric cases have been considered
in the field of formal Laurent series. (1) fix g and choose a random f ∈ L,
(2) fix f and choose a random g∈ L.
In [2], Fuchs proved the following result for case (1).
Theorem 1.8 (M. Fuchs [2]). For almost all f ∈ L, the number of solutions of
(1.4) with 0≤ deg(Q) ≤ N satisfies
Ψ(N ) +O(Ψ(N )1/2(log Ψ(N ))2+ϵ), where ϵ > 0 is an arbitrary constant.
Moreover, Fuchs also obtained generalizations of the above result in [2]. On the other hand, in this research, we will concerned with the problem of (1.4) for case (2). We will obtain a necessary and sufficient condition such that (1.4) has infinitely many solutions. Moreover, we also generalize this result to simultaneous Diophantine approximation. Therefore, we will discuss simultaneous Diophantine approximation next.
Let r, s be positive integers. We denote by Fq[X]r the r-fold Cartesian
product of Fq[X]. Moreover, we denote by Fq(X)r and Fq((X−1))r the vector
spaces over Fq(X) and Fq((X−1)), respectively. Let f = [f1, f2, . . . , fr] be an
element of Fq((X−1))r. Then,
deg(f) = max
j=1,...,rdeg(fj), and deg(0) =−∞.
Define∥ · ∥ a norm with domain Fq((X−1))r and range R+∪ {0} such that ∥f∥ =
qdeg(f). In the sequel, the following sets will be of importance Lr ={f ∈ F
q
((
X−1))r :∥f∥ < 1} and
Lr×s={matrix A of size r × s : all the elements in A are belonging to L} .
We equip Lr with the r-fold product measure of L which we also denote by m.
Now, we consider a Diophantine inequality
where ψ is a {qt: t∈ Z ≥0} → {qt : t∈ Z} function, and q = Q1 Q2 .. . Qr | ∈ Fq[X]r, A = f11 f12 · · · f1s f21 f22 · · · f2s .. . ... . .. ... fr1 fr2 · · · frs ∈ L r×s, g = g1 g2 .. . gs | ∈ Ls.
In fact, the inequality (1.5) is equivalent to the following system of inequalities
|{Q1f11+ Q2f21+· · · + Qrfr1} − g1| < ψ(∥q∥)
|{Q1f12+ Q2f22+· · · + Qrfr2} − g2| < ψ(∥q∥)
.. .
|{Q1f1s+ Q2f2s+· · · + Qrfrs} − gs| < ψ(∥q∥).
will be again concerned with the question of the existence of infinitely many solu-tions to (1.5) as well as the asymptotic number of solusolu-tions to the equation as∥q∥ grows. This subarea is called metric inhomogeneous simultaneous Diophantine
approximation. In fact, we say that (1.5) is in dimension one if r = s = 1, and in
higher dimension if r or s is more than 1. Similar to the result in dimension one, Kristensen investigated the problem of (1.5) if A and g are chosen randomly. Let
Wr,s(ψ) ={(A, g) ∈ Lr×s× Ls : (1.5) has infinitely many solutions q∈ Fq[X]r}
Then, Kristensen proved the following result in [6].
Theorem 1.9 (S. Kristensen [6]). Let ψ : {qt : t ∈ Z
≥0} → {qt : t ∈ Z} be
non-increasing. Then, we have
m(Wr,s(ψ)) = 0, if ∑q∈F q[X]rψ(∥q∥) s <∞, 1, if ∑q∈F q[X]rψ(∥q∥) s =∞. Moreover, let Ψ(N ) := ∑ ∥q∥≤qN ψ(∥q∥)s.
Then, in [6], Kristensen obtained the following result for the number of solutions of (1.5).
Theorem 1.10 (S. Kristensen [6]). Let ψ : {qt : t ∈ Z
≥0} → {qt : t ∈ Z} be
non-increasing. Then, for almost all (A, g)∈ Lr×s× Ls, the number of solutions
of (1.5) with 0 ≤ deg(q) ≤ N satisfies
Ψ(N ) +O ( Ψ(N )12 ( log Ψ(N )32+ϵ )) ,
where ϵ > 0 is an arbitrary constant.
This situation is again called the double-metric case. Moreover, the follow-ing two sfollow-ingle-metric cases are considered in simultaneous Diophantine approxi-mation.
(1) fix g∈ Ls and choose a random A∈ Lr×s;
(2) fix A∈ Lr×s and choose a random g∈ Ls.
In [6], Kristensen also investigated the problem of (1.5) for case (1). Let
Wr×s(ψ, g) :={A ∈ Lr×s : (1.5) has infinitely many solutions q∈ Fq[X]r}.
Here, Kristensen obtained the following result.
Theorem 1.11 (S. Kristensen [6]). Let r ≥ 2 and let ψ : {qt: t ∈ Z
≥0} → {qt :
t ∈ Z} be non-increasing. Then, for any g ∈ Ls,
m (Wr×s(ψ, g)) = 0, if ∑q∈F q[X]rψ(∥q∥) s <∞, 1, if ∑q∈F q[X]rψ(∥q∥) s =∞.
Moreover, Kristensen also obtained a result for the number of solutions of (1.5) in case (1) as well.
Theorem 1.12 (S. Kristensen [6]). Let r ≥ 2, and ψ : {qt: t∈ Z
≥0} → {qt : t∈
Z} be non-increasing. Then, for almost all A ∈ Lr×s, the number of solutions of
(1.5) with 0≤ deg(q) ≤ N satisfies
Ψ(N ) +O ( Ψ(N )12 (log Ψ(N )) 3 2+ϵ ) ,
Note that Kristensen’s result holds for the number of solutions to (1.5) in case (1) as r ≥ 2, and Fuchs obtained the result for the remaining case of r = 1 in [2]. Therefore, this subarea is complete. On the other hand, we will be concerned with the question of the existence of infinitely many solutions to (1.5) in case (2). More specifically, we will find a necessary and sufficient condition such that (1.5) has infinitely many solutions. In the next section, we will introduce the main results of this research.
1.3
Kurzweil’s Theorem in the Field of Formal
Laurent Series
We consider the Diophantine inequality
|{Qf} − g| < 1
qn+ln, Q∈ Fq[X], n = deg(Q), (1.6)
where f is fixed and g is chosen randomly in L. Set
W (ln, f ) :={g ∈ L : (1.6) has infinitely many solutions} .
By the Borel-Cantelli lemma, we obtain that∑n 1
qln <∞ implies m (W (ln, f )) =
0. However, the other direction, namely, m (W (ln, f )) = 1 if
∑
n
1
qln =∞ is not
necessarily true for all sequence ln. Consequently, an important question is as
follows: for which f is m (W (ln, f )) = 1 or 0 according to
∑
nq−l
n converges or
not? In [5], Kim and Nakada obtained a characterization of these f . In order to state their result, define
S := { f ∈ L : ∀ln with ∑ n 1 qln =∞,
(1.6) has infinitely many solutions for almost all g }
.
Moreover, we need the following notation.
Definition 1.1. f is called badly approximable if there exists a constant c > 0
such that for all Q ∈ Fq[X], Q̸= 0 with n = deg(Q),
|{Qf}| > 1 qn+c.
Then, Kim and Nakada proved the following result in [5].
Theorem 1.13 (D. H. Kim and H. Nakada [5]). We have,
S = {f ∈ L : f is badly approximable} .
As for the method of proof, Kim and Nakada used continued fraction ex-pansion in Fq((X−1)). Thus, their method cannot be extended to simultaneous
Diophantine approximation. Here, we will reprove their result with a method closer to the one of Kurzweil who proved the analogue of the above result in the real number field. This new approach not only works in dimension one but also works in higher dimension. Therefore, we consider the Diophantine inequality
∥{qA} − g∥ < 1 q⌊nrs⌋+ln
, q∈ Fq[X]r, n = deg(q), (1.7)
where A is fixed and g is chosen randomly in Ls. Let us again define a set by
Sr×s := { A∈ Lr×s:∀ln with ∑ n 1 qsln =∞,
(1.7) has infinitely many solutions for almost all g∈ Ls }
.
Moreover, similar as above, we need the following notation.
Definition 1.2. A ∈ Lr×s is called badly approximable if there exists a
con-stant c > 0 such that for all q∈ Fq[X]r, q̸= 0 with deg(q) = n,
∥{qA}∥ > 1 q⌊nrs ⌋+c
.
Then, the main result in this research is the following theorem.
Theorem 1.14. We have,
Sr×s=
{
A ∈ Lr×s: A is badly approximable}.
In the next chapter, we will introduce some properties in the field of formal Laurent series which we will use in the proof. Then, we are going to prove Theorem 1.13 in Chapter 3, and Theorem 1.14 in Chapter 4.
Chapter 2
Preliminaries
In this chapter, we will collect some results that we are going to use.
2.1
Fundamental Properties in Dimension One
We start by recalling some results which were already briefly mentioned in the introduction. First, we have the following property (see [8] for a proof).
Proposition 2.1.1. (Fq((X−1)) , +,·) is a field.
Next, |·| is an ultra-metric norm.
Proposition 2.1.2. Let f, g ∈ Fq((X−1)), then |·| satisfies the following:
(1) |f| = 0 ⇔ f = 0. (2) |fg| = |f| |g|.
(3) |f + g| ≤ max {|f| , |g|}. Proof.
(1) |f| = 0 ⇔ deg(f) = −∞ ⇔ f = 0.
(2) |fg| = qdeg(f g)= qdeg(f )+deg(g) = qdeg(f )qdeg(g) =|f| |g|.
(3) |f + g| = qdeg(f +g) = qmax{deg(f),deg(g)}= max{qdeg(f ), qdeg(g)}
Next, recall
L ={f ∈ Fq((X−1)) :|f| < 1
}
which we have equipped with the normalized Haar measure m. For all g ∈ L,
d≥ 1, we define B ( g, 1 qd ) = { f ∈ L : |f − g| < 1 qd } .
Then, we have the following important properties.
Proposition 2.1.3. Each two balls in L are either disjoint or one is contained
in the other.
Proof. Let B(f, q−d), B (g, q−e) be two balls with centers f , g and radii q−d, q−e. Without loss of generality, we suppose d > e. Assume that they are not disjoint, then we have to prove that one is contained in the other. First, we estimate the distance of the two centers f and g. Let h be in the intersection of B(f, q−d)
and B (g, q−e). Then,
|f − g| = |f − h + h − g| ≤ max {|f − h| , |h − g|} < 1 qe.
This means that f ∈ B (g, q−e). Next, we claim that B(f, q−d) belongs to
B (g, q−e). Assume that this is wrong. Then, there exists h in B(f, q−d) \ B (g, q−e). Now,
|h − g| = |h − f + f − g| ≤ max {|h − f| , |f − g|} < 1 qe.
This implies that h ∈ B (g, q−e), a contradiction. Hence, we obtain B(f, q−d)⊆ B (g, q−e), which means that one is contained in the other.
Proposition 2.1.4. Fix b1, b2, . . . , bd∈ Fq, g ∈ L and d ≥ 1. Then, we have
m ({ f : f = d ∑ i=1 biX−i+ ∞ ∑ i=d+1 aiX−i,∀ai ∈ Fq }) = 1 qd, and m ( B ( g, 1 qd )) = 1 qd.
Proof. Assume that h = b1X−1+ b−2X−2+· · · + bdX−d. Thus, f =
∑∞
i=1aiX−i ∈
B(h, q−d) if and only if ai = bi for all i with 1≤ i ≤ d. Consequently,
{ f : f = d ∑ i=1 biX−i+ ∞ ∑ i=d+1 aiX−i,∀ai ∈ Fq } = B ( h, 1 qd ) .
Next, observe that 1 = m (L) = m ∪ c1,...,cd∈Fq { f : f = d ∑ i=1 ciX−i+ ∞ ∑ i=d+1 aiX−i,∀ai ∈ Fq } = ∑ c1,...,cd∈Fq m ({ f : f = d ∑ i=1 ciX−i+ ∞ ∑ i=d+1 aiX−i,∀ai ∈ Fq }) = qdm ({ f : f = d ∑ i=1 ciX−i+ ∞ ∑ i=d+1 aiX−i,∀ai ∈ Fq }) . Hence, m ( B ( h, 1 qd )) = m ({ f : f = d ∑ i=1 biX−i+ ∞ ∑ i=d+1 aiX−i,∀ai ∈ Fq }) = 1 qd
which proves the first result. Since m is a translation-invariant measure, we have that for any g ∈ L, the measures of B(h, q−d) and B(g, q−d) are the same. So, we get
m((g, q−d)) = q−d for any g ∈ L.
We conclude this subsection by recalling Dirichlet’s theorem and providing a proof.
Theorem 2.1 (Analogue of Dirichlet’s Theorem for Formal Laurent series). We
have that
|{Qf}| < |Q|1 , Q∈ Fq[X] (2.1)
Proof. Note that the claimed result is trivial if f is not irrational. Therefore, we
can assume that f is irrational. Now, we need to prove the following claim: for all N ∈ N, there exists a non-zero polynomial Q with deg(Q) ≤ N such that
|{Qf}| < 1
qN. (2.2)
First, we know that the number of Q ̸= 0 with deg(Q) ≤ N is qN +1− 1. We
divide L into qN balls such that
L = ∪ b1,...,bN∈Fq { f : f = N ∑ i=1 biX−i+ ∞ ∑ i=N +1 aiX−i,∀ai ∈ Fq } .
Then there exist at least two different nonzero Q1,Q2 ∈ Fq[X] with deg(Q1), deg(Q2)
≤ N such that |{Q1f} − {Q2f}| < q−N(if not, then the number of Q ̸= 0 with
deg(Q)≤ N is at most qN which is a contradiction). Hence,
1
qN >|{Q1f} − {Q2f}| = |{(Q1− Q2)f}|.
So, (Q1− Q2) is a solution of (2.2). This proves our claim. Moreover, our claim
clearly implies that (2.1) has infinitely many solutions.
2.2 Fundamental Properties in Higher
Dimen-sion
In this section, we will show that all properties from the previous section hold in higher dimension as well.
Let us fix positive numbers r and s. Then, the norm ∥ · ∥ on Fq((X−1))r
from the introduction has the following properties.
Proposition 2.2.1. Let f, g∈ Fq((X−1)) r
, then ∥ · ∥ satisfies the following: (1) ∥f∥ = 0 ⇔ f = 0.
(2) ∥f + g∥ ≤ max {∥f∥, ∥g∥}. Proof.
(1) ∥f∥ = 0 ⇔ deg(f) = −∞ ⇔ f = 0.
(2) ∥f + g∥ = qdeg(f+g)= qmax{deg(f1+g1),deg(f2+g2),...,deg(fr+gr)}
≤ qmax{deg(f1),...,deg(fr),deg(g1),...,deg(gr)}
= qmax{deg(f),deg(g)} = max{∥f∥, ∥g∥}. Recall Lr ={f∈ F q((X−1))r :∥f∥ < 1 }
which we have equipped with the product measure of L (also denoted by m). Moreover, as before, for all g = [g1, . . . , gr]∈ Lr, d≥ 1, we define
B ( g, 1 qd ) = { f ∈ Lr:∥f − g∥ < 1 qd } = r ∏ i=1 B ( gi, 1 qd ) .
As in the one-dimensional case, we again have the following important properties.
Proposition 2.2.2. Each two balls in Lr are either disjoint or one is contained
in the other.
Proof. Let B(f, q−d), B (g, q−e) be two balls with centers f = [f1, . . . , fr], g =
[g1, . . . , gr] and radii q−d, q−e. Without loss of generality, we suppose d > e.
Assume that they are not disjoint. We know that
B ( f, 1 qd ) = r ∏ i=1 B ( fi, 1 qd ) and B ( g, 1 qe ) = r ∏ i=1 B ( gi, 1 qe ) .
Then, by Proposition 2.1.3, we have
B ( fi, 1 qd ) ⊆ B ( gi, 1 qe ) , ∀i = 1, . . . , r.
This implies that
r ∏ i=1 B ( fi, 1 qd ) ⊆ r ∏ i=1 B ( gi, 1 qe ) .
Hence, we obtain B(f, q−d) ⊆ B (g, q−e) which means that one is contained in the other.
Proposition 2.2.3. Let d be a positive integer. Fix b(j)i ∈ Fq for all i = 1, . . . , d
and j = 1, . . . , r. Then, we have m ({ f = [f1, . . . , fr] : fj = d ∑ i=1 b(j)i X−i+ ∞ ∑ i=d+1 a(j)i X−i,∀a(j)i ∈ Fq,∀j }) = 1 qrd and m ( B ( g, 1 qd )) = 1 qrd.
Proof. Assume that h = [h1, . . . , hr], where hj =
∑d i=1b
(j)
i X−i for all j. Then, for
any f = [f1, . . . fr] with fj = ∑∞ i=1a (j) i X−i, f∈ B ( h, 1 qd ) iff a(j)i = b(j)i , 1≤ i ≤ d, ∀j.
This implies that { f = [f1, . . . , fs] : fj = d ∑ i=1 b(j)i X−i+ ∞ ∑ i=d+1 a(j)i X−i,∀a(j)i ∈ Fq,∀j } = B ( h, 1 qd ) . Consequently, m ({ f = [f1, . . . , fs] : fj = d ∑ i=1 b(j)i X−i+ ∞ ∑ i=d+1 a(j)i X−i,∀a(j)i ∈ Fq,∀j }) = m ( B ( h, 1 qd )) = m ( r ∏ i=1 B ( hi, 1 qd )) = r ∏ i=1 m ( B ( hi, 1 qd )) = r ∏ i=1 1 qd = 1 qrd.
Since m is a translation invariant measure, we have
m ( B ( h, 1 qd )) = m ( B ( g, 1 qd ))
for any g∈ Lr. So, we get
m ( B ( g, 1 qd )) = 1 qrd for any g ∈ L r.
Next, we need the following notation.
Definition 2.1. A r× s matrix A is called irrational if qA does not belong to
As in dimension one, we conclude by stating and proving Dirichlet’s theo-rem.
Theorem 2.2 (Analogue of Dirichlet’s Theorem for Formal Laurent Series). We
have that
∥{qA}∥ < 1 q⌊nrs⌋
, q∈ Fq[X]r, deg(q) = n (2.3)
has infinitely many solutions.
Proof. Note that the claimed result is trivial if A is not irrational. Therefore, we
can assume that A is irrational. Then, similar as in the one-dimensional case, we need to prove the following claim: for all N ∈ N, there exists a non-zero polynomial vector q with deg(q)≤ N such that
∥{qA}∥ < 1 q⌊N rs ⌋
(2.4) First, we know that the number of q ̸= 0 with deg(q) ≤ N is q(N +1)r − 1.
We divide L into q⌊N rs ⌋ balls as in the proof of Dirichlet’s theorem in dimension
one. This yields that a subdivision of Ls into q⌊N rs ⌋s balls. Then, there are two
different nonzero polynomial vectors q1,q2 with deg(q1), deg(q2) ≤ N such that
∥{q1A} − {q2A}∥ < q−⌊
N r s ⌋
(if not, then the number of q with deg(q)≤ N is at most q⌊N rs ⌋s ≤ q N r s s= qN r, a contradiction). Hence, 1 q⌊N rs ⌋ >∥{q1A} − {q2A}∥ = ∥{(q1− q2)A}∥
So, (q1− q2) is a solution of (2.4). This proves our claim. Moreover, due to the irrationality of A, our claim implies that (2.3) has infinitely many solutions.
Chapter 3
Kurzweil’s Theorem in
Dimension One
Here, we are going to prove Theorem 1.13 from the introduction. Therefore, fix a f = f1X−1+ f2X−2+· · · . For the next three lemmas, we assume that f is
irrational.
Lemma 3.1. {{Qf} : Q ∈ Fq[X]} is dense in L.
Proof. Let us fix n ∈ N and g = g1X−1+ g2X−2+· · · , where gi ∈ Fq. Then we
claim: there exists Q with deg(Q) = N such that |{Qf} − g| < q−n. In order to prove this, we consider g = aA, where
g = g1 g2 ... gn | , a = a0 a1 ... aN | , A = f1 f2 · · · fn f2 f3 · · · fn+1 ... ... . .. ... fN +1 fN +2 · · · fN +n .
Then we claim: rank(A) = n as N is large enough. We need to prove that the column vectors of A are linear independent. Suppose that this is wrong. Then,
α1(f1, f2, . . . , fN +1) +· · · + αn(fn, fn+1, . . . , fN +n) = (0, 0, . . . , 0)
with αi not all zero. Let P (X) = α1 + α2X +· · · + αnXn−1, P (X) ̸= 0. Then,
fixed), which means that min|{P f}| ≥ q−N−1 for N large enough. This gives a contradiction. Therefore, P (X) must be 0, which implies that the column vectors of A are linear independent. Then for all g, there exists a such that aA = g. This implies that Q = a0+ a1X +· · ·+aNXN satisfies|{Qf} − g| < q−n. Hence,
{{Qf} : Q ∈ Fq[X]} is dense in L.
Lemma 3.2 (0-1 law). Let a measurable set E inL be invariant under the action
· + {Qf} for all Q ∈ Fq[X]. Then, we have m (E) = 0 or 1.
Proof. Suppose that m (E) > 0. By [3], for all ϵ > 0, there exists a radius q−d
such that ∫ χE(g)− m ( E∩ ( B ( g,q1d ) +{Qf} )) m ( B ( g,q1d ) +{Qf} ) dm < ϵ m(E)
for all Q∈ Fq[X]. Consequently,
∫ E 1 −m ( E∩ ( B ( g, 1 qd ) +{Qf} )) m ( B ( g,q1d ) +{Qf} ) dm < ϵ m(E). This implies that there exists a g ∈ L with
1− m ( E∩ ( B ( g,q1d ) +{Qf} )) m ( B ( g,q1d ) +{Qf} ) < ϵ. Thus m ( E∩ ( B ( g,q1d ) +{Qf} )) m ( B ( g,q1d ) +{Qf} ) > 1− ϵ.
Since {{Qf} : Q ∈ Fq[X]} is dense in L, we get the inequality
m (E) > 1− ϵ
for all ϵ > 0. Hence, we obtain the result m (E) = 1.
Lemma 3.3. Let
E :=
{
g ∈ L : |{Qf} − g| < 1
qn+ln with n = deg(Q) has infinitely many solutions
}
.
Then, E is invariant under the action · + {Qf} for all Q ∈ Fq[X] and hence
Proof. Fix a polynomial Q′. Let g ∈ E. Then, we can find infinitely many Q with deg(Q) > deg(Q′) such that
|{(Q − Q′)f} − g| = |{Qf} − (g + {Q′f})| < 1
qn+ln.
So, we get E +{Q′f} ⊆ E. Conversely, since |{(Q + Q′)f} − g| < q−n−ln has
in-finitely many solutions, we get that |{Qf} − (g − {Q′f})| < q−n−ln has infinitely
many solutions. Thus, g−{Q′f} ∈ E. Then, g = g−{Q′f}+{Q′f} ∈ E +{Q′f},
this means E ⊆ E + {Q′f}. So, we obtain E = E + {Q′f}. Consequently, E is
invariant under the action · + {Qf} for all Q ∈ Fq[X] and hence m(E) = 0 or 1.
For the next two lemmas, f is assumed to be badly approximable. Thus, there exists a constant c > 0 such that for all Q∈ Fq[X], Q̸= 0 with n = deg(Q),
|{Qf}| > 1 qn+c.
Lemma 3.4. Let g ∈ L. Then, the number of {Qf} with deg(Q) ≤ N belonging
to B(g, q−d) is at most max{qN +c−d, 1}.
Proof. First, we need the following claim: define g = g1X−1+g2X−2+· · ·+gdX−d+
· · · , where d > 0. Then, the number of {Qf} with deg(Q) ≤ N belonging to B(g, q−d) is either qσ or 0, where σ ≥ 0. Let a
i be the coefficient of Xi of Q, ∀i = 1, 2, . . . , N. Define a = a0 a1 .. . aN | , b = g1 g2 .. . gd | , A = f1 f2 · · · fd f2 f3 · · · fd+1 .. . ... . .. ... fN +1 fN +2 · · · fN +d ,
Let us consider the linear system
aA = b.
We have to discuss the number of solutions (which is equal to the number of
{Qf} with deg(Q) ≤ N belonging to B(g, q−d)). There are two cases:
1. If the linear system has no solution, then the number of{Qf} which belong to B(g, q−d) is 0.
2. If the linear system has a particular solution a = [a0, a1, . . . , aN], then
a + ker A is the set of all solutions of the linear system. Hence the number
of solutions is | ker A| = qdim ker A = qσ.
By the above cases, we obtain our claim. Next, we suppose qσ is the number of
{Qf} with deg(Q) ≤ N belonging to B(g, q−d). We consider the following cases: 1. If σ = 0, then the number of {Qf} belonging to B(g, q−d) is 1.
2. If σ > 0, then there exist two points {Q1f}, {Q2f} ∈ B
(
g, q−d) such that
|{Q1f} − {Q2f}| < q−d−(σ−1), where deg(Q1), deg(Q2) ≤ N. Because f is
badly approximable, we get 1 qd+(σ−1) >|{Q1f} − {Q2f}| =|{(Q1− Q2)f}| > 1 qdeg(Q1−Q2)+c ≥ 1 qN +c.
This implies that qσ ≤ qN +c−d.
By the two cases, we obtain that the number of{Qf} with deg(Q) ≤ N belonging to B(g, q−d) is at most max{qN +c−d, 1}.
We give a second method of proof which is in fact easier.
Second Method of Proof. Let Q, Q′be two different polynomials with deg(Q), deg(Q′)
≤ N. Because f is badly approximable, we have |{Qf} − {Q′f}| = |{(Q − Q′)f}| > 1
qdeg(Q−Q′)+c ≥
1
qN +c.
This means that the distance between two points {Qf}, {Q′f} with deg(Q),
deg(Q′)≤ N is at least q−N−c. Now, we consider two cases: 1. If q−N−c ≥ q−d, then there is at most one point in B(g, q−d). 2. If q−N−c < q−d, then the number of points in B(g, q−d) is at most
q−d q−N−c = q
N−d+c .
Hence, the number of {Qf} with deg(Q) ≤ N belonging to B(g, q−d) is at most max{qN−d+c, 1}.
Lemma 3.5. Let ln be a sequence with
∑ 1
qln =∞. Then, for all k ≥ 0, we have
m ∪∞ n=k ∪ deg(Q)=n B ( {Qf}, 1 qn+ln ) > 1 qc+1. (3.1)
Proof. We first exclude the case q = 2.
Let ln′ = max{ln, c}, ∀n ∈ N. Then, we have
∑
q−l′n =∞. Assume that (3.1) is
false. Hence, there exists k0 ∈ N such that
m ∪N n=k0 ∪ deg(Q)=n B ( {Qf}, 1 qn+l′n ) ≤ 1 qc+1 , for all N ≥ k0. (3.2) We define a set LN := deg(Q) = N :{Qf} ∈ N ∪ n=k0 ∪ deg(Q′)=n B ( {Q′f} , 1 qn+l′n ) \ N∪−1 n=k0 ∪ deg(Q′)=n B ( {Q′f} , 1 qn+l′n ) . We first estimate the number of elements of LN. Let
N∪−1 n=k0 ∪ deg(Q′)=n B ( {Q′f} , 1 qn+l′n ) =∪ i B ( {Qif} , 1 qdi ) , where B({Qif} ,q1di )
are disjoint,∀i. By (3.2), we get 1 qc+1 ≥ m N∪−1 n=k0 ∪ deg(Q′)=n B ( {Q′f}, 1 qn+l′n ) = m ( ∪ i B ( {Qif} , 1 qdi )) =∑ i m ( B ( {Qif} , 1 qdi )) =∑ i 1 qdi.
Using Lemma 3.4, the number of Q with deg(Q)≤ N such that {Qf} belong to ∪
iB
(
{Qif},q1di
)
is at most ∑imax{qN +c−di, 1} = max{qN +c∑
iq−di, q N} =
qN. Thus, the number of L N is at least qN +1− qN − qN = qN(q− 2). Next, we claim that ∪ Q∈LN B ( {Qf}, 1 qN +lN′ ) ⊂ N ∪ n=k0 ∪ deg(Q′)=n B ( {Q′f}, 1 qn+l′n ) \ N∪−1 n=k0 ∪ deg(Q′)=n B ( {Q′f}, 1 qn+l′n ) . (3.3) In order to show this, fix Q1 ∈ LN. Suppose there exists a polynomial Q2 with
deg(Q2) = u < N and B
(
{Q1f}, q−N−l
′
N)∩ B({Q2f}, q−u−lu′) ̸= ∅. We know
that {Q1f} does not belong to B
( {Q2f}, q−u−l ′ u). Hence, B ( {Q2f}, 1 qu+l′u ) ⊂ B ( {Q1f}, 1 qN +l′N ) . Then, we get |{Q1f} − {Q2f}| < 1 qN +l′N.
By Lemma 3.4, the number of {Qf} belonging to B({Q1f}, q−N−l
′
N) is at most
max{qN−N−l′N+c, 1} = max{q−l′N+c, 1} = 1. Thus, we get {Q
1f} = {Q2f}, a
contradiction. Consequently, (3.3) holds. Now, we show that any two balls ap-pearing on the left side of (3.3) are disjoint. We again use proof by contradiction. Therefore, suppose there are two different polynomials Q1, Q2 ∈ LN such that
B({Q1f}, q−N−l
′
N)and B({Q2f}, q−N−l′N) are not disjoint. Thus, we know that
these two balls are equal. This implies that
|{Q1f} − {Q2f}| = |{(Q1− Q2)f}| < 1 qN +lN′ . Hence, {(Q1 − Q2)f} ∈ B ( 0, q−N−l′N ) .
By Lemma 3.4 again, the number of {Qf} in B({Q1f}, q−N−l
′
N) is at most
max{qc−l′N, 1} = 1. Consequently, {Q
latter claim and (3.3), we now obtain m ∪N n=k0 ∪ deg(Q)=n B ( {Qf}, 1 qn+l′n ) ≥ m N∪−1 n=k0 ∪ deg(Q)=n B ( {Qf}, 1 qn+l′n ) + m ( ∪ Q∈LN B ( {Qf}, 1 qN +l′N )) ≥ m N−1∪ n=k0 ∪ deg(Q)=n B ( {Qf}, 1 qn+l′n ) + (q − 2)qN 1 qN +l′N ≥ m N∪−2 n=k0 ∪ deg(Q)=n B ( {Qf}, 1 qn+l′n ) + q − 2 qlN′ −1 + q− 2 ql′N ≥ · · · ≥ (q − 2) N ∑ n=k0 1 ql′n .
As the series ∑∞n=1q−l′n diverges, we have a contradiction for N large enough.
Now, we consider the case q = 2. Since ∑n≥0q−l′n = ∞, we have either
∑ n≥0q−l ′ 2n = ∞ or ∑ n≥0q−l ′
2n+1 = ∞. Without loss of generality, assume that
the first case holds. Then, the same proof as above can be used with the one difference that is instead of LN, we consider
L2N := { deg(Q) = 2N :{Qf} ∈ 2N ∪ n=k0 ∪ deg(Q′)=n B ( {Q′f} , 1 qn+l′n ) \ 2N∪−2 n=k0 ∪ deg(Q′)=n B ( {Q′f} , 1 qn+l′n ) . Hence, we obtain m ∪2N n=k0 ∪ deg(Q)=n B ( {Qf}, 1 qn+l′n ) ≥ d ∑N n=⌈k02 ⌉ q−l′2n
for some d > 0. As the series ∑∞n=1q−l′2n =∞, we have a contradiction again for
N large enough.
Proposition 3.1.
Proof. Let f be badly approximable. We have to show that m ∩∞ k=0 ∞ ∪ n=k ∪ deg(Q)=n B ( {Qf}, 1 qn+ln ) = 1. By Lemma 3.5, we obtain m ∪∞ n=k ∪ deg(Q)=n B ( {Qf}, 1 qn+ln ) > 1 qc+1 > 0, ∀k. Consequently, m ∩∞ k=0 ∞ ∪ n=k ∪ deg(Q)=n B ( {Qf}, 1 qn+ln ) > 0 and Lemma 3.3 implies the claim.
Proposition 3.2.
S ⊆ {f ∈ L : f is badly approximable}
Proof. Assume that f is not badly approximable. We will show that a sequence ln we can choose such that
∑∞
n=1
1
qln =∞ but for almost every g ∈ L, there are
at most finitely many Q with
|{Qf} − g| < 1
qn+ln, Q∈ Fq[X], deg(Q) = n.
Let us choose (Ri, Si) such that
f − Ri
Si
≤ q2n1i+2i,∀i ∈ N,
where deg(Si) = ni, and define
t0 = 0 ti = ni+ i, i≥ 1 , and for ti−1 ≤ n < ti, ln= ti− n . Then, we have ∞ ∑ n=1 1 qln ≥ ∞ ∑ i=1 1 qlti−1 = ∞ ∑ i=1 1 qti−(ti−1) = ∞ ∑ i=1 1 q =∞.
On the other hand, let Q be a polynomial such that deg(Q) < ti, ∀i. Then, Qf − QRi Si ≤ qti q2ni+2i.
This implies that
{Qf} − R′i
Si
≤ qn1i+i < 1.
Note that deg(R′i) < deg(Si). Therefore,
∪ ti−1≤n<ti ∪ deg(Q)=n B ( {Qf}, 1 qn+ln ) = ∪ ti−1≤n<ti ∪ deg(Q)=n B ( {Qf}, 1 qti ) ⊂ ∪ deg(R′i)<deg(Si) B ( R′i Si , 1 qti ) .
Then, we can estimate the measure of union of these balls
m ∪ ti−1≤n<ti ∪ deg(Q)=n B ( {Qf}, 1 qn+ln ) ≤ m ∪ deg(Ri)<deg(Si) B ( R′i Si , 1 qti ) ≤ |Si| qti = qni qti = 1 qi. So we get ∞ ∑ i=1 m ∪ ti−1≤n<ti ∪ deg(Q)=n B ( {Qf}, 1 qn+ln ) ≤∑∞ i=1 1 qi <∞.
Hence, for almost every g ∈ L, there are at most finitely many Q′s such that |{Qf} − g| < q−n−ln with deg(Q) = n.
Chapter 4
Kurzweil’s Theorem in Higher
Dimension
Here, we are going to prove Theorem 1.14 from the introduction. Therefore, fix a r× s matrix A. We first need a technical lemma.
Lemma 4.1. If Au| ∈ Fq[X]r for some u|̸= 0, then A is not badly approximable.
Proof. Assume that A is badly approximable. Let us fix some notation. First, set
A = f11 f12 · · · f1s f21 f22 · · · f2s ... ... ... ... fr1 fr2 · · · frs , u| = U1 U2 ... Us and Au|= f11 f12 · · · f1s f21 f22 · · · f2s .. . ... . .. ... fr1 fr2 · · · frs U1 U2 .. . Us = U1f11+· · · + Usf1s U1f21+· · · + Usf2s .. . U1fr1+· · · + Usfrs ≡ R1 R2 .. . Rr
which is in Fq[X]r. Then, by Dirichlet’s theorem, |Q1f11+ Q2f21+· · · + Qrfr1− P1| < q−⌊ N r s−1⌋ |Q1f12+ Q2f22+· · · + Qrfr2− P2| < q−⌊ N r s−1⌋ ... |Q1f1s−1+ Q2f2s−1+· · · + Qrfrs−1− Ps−1| < q−⌊ N r s−1⌋
has infinitely many solutions in Q1, Q2, . . . , Qr and P1, P2, . . . , Ps−1 with N =
max1≤i≤rdeg(Qi). Next, multiply both sides of the above inequality by |Us| and
set Q′i := UsQi and Pj′ := UsPj for all i, j. Then, we obtain
|Q′ 1f11+ Q′2f21+· · · + Q′rfr1− P1′| < |Us|q−⌊ N r s−1⌋ |Q′ 1f12+ Q′2f22+· · · + Q′rfr2− P2′| < |Us|q−⌊ N r s−1⌋ ... |Q′ 1f1s−1+ Q′2f2s−1+· · · + Q′rfrs−1− Ps′−1| < |Us|q−⌊ N r s−1⌋.
This implies that
|Q′ 1f11+ Q′2f21+· · · + Q′rfr1 − P1′| < q−⌊ N ′r s−1⌋−c1 |Q′ 1f12+ Q′2f22+· · · + Q′rfr2 − P2′| < q−⌊ N ′r s−1⌋−c1 ... |Q′ 1f1s−1+ Q′2f2s−1+· · · + Q′rfrs−1− Ps′−1| < q−⌊ N ′r s−1⌋−c1 (4.1)
has infinitely many solutions in Q′1, . . . , Q′r and P1′, . . . , Pr′, where
N′ = maxi=1,...,rdeg(Q′i) and c1 is a suitable constant. Now, consider
Usf1sQ′1+· · · + UsfrsQ′r = r ∑ i=1 (Ri− U1fi1− · · · − Us−1fis−1) Q′i = r ∑ i=1 Q′iRi− s−1 ∑ j=1 Uj ( Q′1f1j +· · · + Q′rfrj − Pj′ ) − s−1 ∑ j=1 UjPj′
This implies that
r ∑ i=1 UsfisQ′i+ s−1 ∑ j=1 UjPj′− r ∑ i=1 Q′iRi =− s−1 ∑ j=1 Uj ( Q′1f1j +· · · + Q′rfrj − Pj′ )
Hence, r ∑ i=1 UsfisQ′i+ s−1 ∑ j=1 UjPj′ − r ∑ i=1 Q′iRi ≤ max j=1,...,s−1 { |Uj| Q′1f1j +· · · + Q′rfrj− Pj′ } < q−⌊ N ′r s−1⌋−c2,
where c2 is a suitable constant. Dividing both sides by|Us| gives
r ∑ i=1 fisQ′i+ ∑s j=1UjPj′− ∑r i=1Q′iRi Us < q−⌊ N ′r s−1⌋−c3,
where c3 is a suitable constant. Since Us divides Q′i and Pj′ for all i, j, we obtain
T = ∑s j=1UjPj′− ∑r i=1Q′iRi Us
is a polynomial. Thus, we have proved that
|Q′
1f1s+· · · + Q′rfrs+ T| < q−⌊
N r s−1⌋−c3.
Now, set q′ = [Q′1, Q′2, . . . , Q′r] and c = min{c1, c3}. By the above inequality and
(4.1),
∥{q′A}∥ < q−⌊N ′r s−1⌋−c
has infinitely many solutions. Consequently, we obtain that A is not badly ap-proximable, a contradiction. Hence, the proof is finished.
For the next five lemmas, we assume that A = [fij]r×s is badly
approx-imable. Then, there exists a constant c > 0 such that for all q ∈ Fq[X]r, q̸= 0
with deg(q) = n,
∥{qA}∥ > 1 q⌊nrs ⌋+c
.
Lemma 4.2. {{qA} : q ∈ Fq[X]r} is dense in Ls.
Proof. Let us fix n ∈ N and g = [g1, g2, . . . , gs]∈ Ls, where
gj = g
(1)
j X−1+ g
(2)
j X−2+· · · , ∀j = 1, 2, . . . , s.
We have to show that there exists q = [Q1, Q2, . . . , Qr] with deg(Qi) = Ni such
that
∥{qA} − g∥ < 1
First, for i = 1, 2, . . . , r and j = 1, 2, . . . , s, set fij = f (1) ij X−1+ f (2) ij X−2+· · · , and let ai = a(0)i a(1)i ... a(N )i | , Aij = fij(1) fij(2) · · · fij(n) fij(2) fij(3) · · · fij(n+1) ... ... . .. ... fij(N +1) fij(N +2) · · · fij(N +n) , bj = gj(1) gj(2) ... g(n)j | . Finally, set a = a1 a2 .. . ar | , A′ = A11 A12 · · · A1s A21 A22 · · · A2s .. . ... . .. ... Ar1 Ar2 · · · Ars , b = b1 b2 .. . bs | .
Then, the inequality (4.2) has a solution if and only if aA′ = b has a solution a. In order to prove that this system is solvable, we have to show that rank(A′) = sn as N is large enough. Assume that there exist α1, . . . , αsn not all zero such that
α1(f (1) 11 , . . . , f (N +1) 11 , . . . , f (1) r1 , . . . , f (N +1) r1 ) + . . . + αsn(f (n) 1s , . . . , f (N +n) 1s , . . . , f (n) rs , . . . , frs(N +n)) = 0, (4.3) Now, we set p = [P1, P2, . . . , Ps]∈ Fq[X]s, p̸= 0 with
P1(X) = α1+ α2X +· · · + αnXn−1
P2(X) = αn+1+ αn+2X +· · · + α2nXn−1
...
Ps(X) = α(s−1)n+1+ α(s−1)n+2X +· · · + αsnXn−1.
Hence, (4.3) can be rewritten to
| {P1fi1+· · · + Psfis} | < q−N−1
for all i = 1, 2, . . . , r. This implies that
On the other hand, since A is badly approximable, Lemma 4.1 implies that Ap| does not belong toFq[X]r,∀p ̸= 0. Consequently, since the number of p is finite
(since n is fixed),
min
deg(p)<n, p̸=0∥ {Ap
|} ∥ ≥ q−N−1,
for N large enough, a contradiction. Hence, we obtain that p = 0 which implies
α1 = α2 = · · · = αsn = 0. Thus, our claimed result is proved. Therefore, there
exists a solution of aA′ = b. This implies that for all n ∈ N, (4.2) has a solution. Finally, we have proved that {{qA} : q ∈ Fq[X]r} is dense in L.
The next two lemmas are proved as in the last chapter. Consequently, we will omit the proofs.
Lemma 4.3 (0-1 law). Let a measurable set E inLsbe invariant under the action
· + {qA} for all q ∈ Fq[X]r. Then, we have m(E) = 0 or 1.
Lemma 4.4. Let
E :=
{
g :∥ {qA} − g∥ < 1
q⌊nrs⌋+ln
with n = deg(q) has infinitely many solutions
}
.
Then, E is invariant under the action · + {qA} for all q ∈ Fq[X]r and hence
m(E) = 0 or 1.
Next, we need the following result which is similar to Lemma 3.4 from the last chapter.
Lemma 4.5. Let g∈ Ls and d > 0. Then, the number of{qA} with deg(q) ≤ N
belonging to B(g, q−d) is at most max{qN r+cs−ds, 1}.
Proof. For the proof, we use the second method of proof of Lemma 3.4. Therefore,
fix q, q′ ∈ Fq[X]r with deg(q), deg(q′)≤ N. Since A is badly approximable, we
have ∥{qA} − {q′A}∥ = ∥{(q − q′)A}∥ > 1 q⌊deg(qs−q′)r⌋+c ≥ 1 q⌊N rs ⌋+c .
This means that the distance between any two points {qA} and {q′A} is more
than q−⌊N rs ⌋−c. Then, we consider the following two cases.
2. If q−⌊N rs ⌋−c < q−d, then the number of points in B(g, q−d) is at most ( q−d)s ( q−⌊N rs ⌋−c )s ≤ qN r+cs−ds.
Hence, our claim is proved.
Lemma 4.6. Let ln be a sequence with
∑
q−sln = ∞. Then, for all k ≥ 0, we
have m ∪∞ n=k ∪ deg(q)=n B ( {qA}, 1 q⌊nrs⌋+ln ) > 1 qcs+1. (4.4)
Proof. We first exclude the case q = 2 and r = 1.
Let ln′ = max{ln, c}, ∀n ∈ N. Then, we have
∑
q−sl′n =∞. Assume that (4.4) is
incorrect. Hence, there exists k0 ∈ N such that
m ∪N n=k0 ∪ deg(q)=n B ( {qA}, 1 q⌊nrs⌋+ln′ ) ≤ 1 qcs+1 , for all N ≥ k0. (4.5) We define a set LN = deg(q) = N :{qA} ∈ N ∪ n=k0 ∪ deg(q′)=n B ( {q′A} , 1 q⌊nrs⌋+l′n ) \ N∪−1 n=k0 ∪ deg(q′)=n B ( {q′A} , 1 q⌊nrs ⌋+l′n ) . We first estimate the number of elements of LN. Let
N∪−1 n=k0 ∪ deg(q′)=n B ( {q′A} , 1 q⌊nrs⌋+l′n ) =∪ i B ( {qiA} , 1 qdi ) ,
where B({qiA} , q−di) are disjoint∀i. By (4.5), we get
1 qcs+1 ≥ m N∪−1 n=k0 ∪ deg(q′)=n B ( {q′A}, 1 q⌊nrs ⌋+l′n ) = m ( ∪ i B ( {qiA} , 1 qdi )) =∑ i m ( B ( {qiA} , 1 qdi )) =∑ i 1 qsdi.
