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2008 | Buch

Diophantine Approximation

Festschrift for Wolfgang Schmidt

herausgegeben von: Hans Peter Schlickewei, Klaus Schmidt, Robert F. Tichy

Verlag: Springer Vienna

Buchreihe : Developments in Mathematics

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This volume contains 22 research and survey papers on recent developments in the field of diophantine approximation. The first article by Hans Peter Schlickewei is devoted to the scientific work of Wolfgang Schmidt. Further contributions deal with the subspace theorem and its applications to diophantine equations and to the study of linear recurring sequences. The articles are either in the spirit of more classical diophantine analysis or of geometric or combinatorial flavor. In particular, estimates for the number of solutions of diophantine equations as well as results concerning congruences and polynomials are established. Furthermore, the volume contains transcendence results for special functions and contributions to metric diophantine approximation and to discrepancy theory. The articles are based on lectures given at a conference at the Erwin Schr6dinger Institute in Vienna in 2003, in which many leading experts in the field of diophantine approximation participated. The editors are very grateful to the Erwin Schr6dinger Institute and to the FWF (Austrian Science Fund) for the financial support and they express their particular thanks to Springer-Verlag for the excellent cooperation. Robert E Tichy Diophantine Approximation H. E Schlickewei et al. , Editors 9 Springer-Verlag 2008 THE MATHEMATICAL WORK OF WOLFGANG SCHMIDT Hans Peter Schlickewei Mathematik Informatik, und Philipps-Universitiit Hans-Meerwein-Strasse, Marburg, 35032 Marburg, Germany k. Introduction Wolfgang Schmidt's mathematical activities started more than fifty years ago in 1955. In the meantime he has written more than 180 papers - many of them containing spectacular results and breakthroughs in different areas of number theory.

Inhaltsverzeichnis

Frontmatter
The Mathematical Work of Wolfgang Schmidt
Abstract
Wolfgang Schmidt’s mathematical activities started more than fifty years ago in 1955. In the meantime he has written more than 180 papers – many of them containing spectacular results and breakthroughs in different areas of number theory.
Hans Peter Schlickewei
SchÄffer’s Determinant Argument
Abstract
Let ‖...‖ denote distance from the nearest integer. Various versions of the following problem in simultaneous Diophantine approximation have been studied since 1957, beginning with Danicic [5]. Given an integer h ≥ 2. we seek a number θ having the following property, for every ∈ > 0 and every pair α = (α1, ... αh), β = (β1,..., βh) in ℝh: For N > C(h, ∈, there is an integer n, 1 ≤ nN, satisfying
$$ \left\| {n^2 \alpha j + n\beta j} \right\| < N^{ - \theta + \in } \left( {j = 1, \ldots ,h} \right). $$
Roger C. Baker
Arithmetic Progressions and Tic-Tac-Toe Games
Abstract
This paper is partly an overview of the subject (see Sections 1–4), in fact, as far as I know, the first attempt to do that, and partly an ordinary research paper containing proofs for new results (Sections 5–8). I use many different sources; to make the reader’s life easier, I decided to keep the paper (more-or-less) self-contained - this explains the considerable length.
József Beck
Metric Discrepancy Results for Sequences {nkx} and Diophantine Equations
Abstract
Let (n k ) be an increasing sequence of positive integers. For 0 ≤ x ≤ 1, set
$$ \eta _k = \eta _k \left( x \right): = n_k x \left( {mod 1} \right). $$
(1)
István Berkes, Walter Philipp, Robert F. Tichy
Mahler’s Classification of Numbers Compared with Koksma’s, II
Abstract
Mahler [8], in 1932, and Koksma [7], in 1939, introduced two related measures of the degree of approximation of a complex transcendental number ξ by algebraic numbers. Following Mahler [8], for any integer n ≥ 1, we denote by wn(ξ) the supremum of the exponents w for which
$$ 0 < \left| {P\left( \xi \right)} \right| < H\left( P \right)^{ - \omega } $$
has infinitely many solutions in integer polynomials P(X) of degree at most n. Here, H(P) stands for the naïve height of the polynomial P(X), that is, the maximum of the absolute values of its coefficients. Following Koksma [7], for any integer n ≥ 1, we denote by w n ξ the supremum of the exponents w for which
$$ 0 < \left| {\xi - \alpha } \right| < H\left( \alpha \right)^{ - \omega - 1} $$
has infinitely many solutions in complex algebraic numbers α of degree at most n. Here, H(α) stands for the naïve height of α, that is, the naïve height of its minimal defining polynomial over Z. Clearly, the functions w1 and w 1 * coincide.
Yann Bugeaud
Rational Approximations to A q-Analogue of π and Some Other q-Series
Abstract
One of the famous mathematical constants is π, Archimedes’ constant. There are several analytic ways to define it, e.g., by the (slowly convergent) series
$$ \pi = 4\sum\limits_{v = 0}^\infty {\frac{{\left( { - 1} \right)^v }} {{2^v + 1}},} $$
(1)
or by the (Gaussian probability density) integral
$$ \pi = \left( {\int\limits_{ - \infty }^\infty {e^{ - x^2 } dx} } \right)^2 ; $$
(2)
for a comprehensive exposition of different representations and bibliography we refer the reader to [Fi, Section 1.4].
Peter Bundschuh, Wadim Zudilin
Orthogonality and Digit Shifts in the Classical Mean Squares Problem in Irregularities of Point Distribution
Abstract
Suppose that \( \mathcal{A}_N \) is a distribution of N > 1 points, not necessarily distinct, in the n-dimensional unit cube U n = [0, l) n , where n ≥ 2. We consider the L2-discrepancy
$$ \mathcal{L}_2 \left[ {\mathcal{A}_N } \right] = \left( {\int\limits_{U^n } {\left| {\mathcal{L}\left[ {\mathcal{A}_N ;Y} \right]} \right|} ^2 dY} \right)^{1/2} , $$
where for every Y = (y1,..., y n) ∈ U n , the local discrepancy \( \mathcal{L}\left[ {\mathcal{A}_N ;Y} \right] \) is given by
$$ \mathcal{L}\left[ {\mathcal{A}_N ;Y} \right] = \# \left( {\mathcal{A}_N \cap B_Y } \right) - N vol B_Y . $$
Here
$$ B_Y = \left[ {0,y_1 } \right) \times \ldots \times \left[ {0,y_n } \right) \subseteq U^n $$
is a rectangular box of volume vol By = y1... y n , and #(S) denotes the number of points of a set S, counted with multiplicity.
William W. L. Chen, Maxim M. Skriganov
Applications of the Subspace Theorem to Certain Diophantine Problems
A survey of some recent results
Abstract
One of the cornerstones of modern Diophantine Approximation is the Schmidt Subspace Theorem. Its original form was obtained by Wolfgang Schmidt around 1970, as an evolution of slightly special cases related to an analogue of Roth’s Theorem for simultaneous rational approximations to several algebraic numbers. While Roth’s Theorem considers rational approximations to a given algebraic point on the line, the Subspace Theorem deals with approximations to given hyperplanes in higher dimensional space, defined over the field of algebraic numbers, by means of rational points in that space.
Pietro Corvahja, Umberto Zannier
A Generalization of the Subspace Theorem With Polynomials of Higher Degree
Abstract
1.1 The Subspace Theorem can be stated as follows. Let K be a number field (assumed to be contained in some given algebraic closure https://static-content.springer.com/image/chp%3A10.1007%2F978-3-211-74280-8_9/978-3-211-74280-8_9_IEq1_HTML.gif of ℚ), n a positive integer, 0 < δ 1 and S a finite set of places of K. For v ∈ S, let \( L_0^{\left( v \right)} , \ldots ,L_n^{\left( v \right)} \) be linearly independent linear forms in https://static-content.springer.com/image/chp%3A10.1007%2F978-3-211-74280-8_9/978-3-211-74280-8_9_IEq3_HTML.gif [x 0,...,x n ]. Then the set of solutions x ∈ℙn(K) of
$$ \log \left( {\prod\limits_{v \in S} {\prod\limits_{i = 0}^n {\frac{{\left| {L_i^{\left( v \right)} \left( x \right)} \right|_v }} {{\left\| x \right\|_v }}} } } \right) \leqslant - \left( {n + 1 + \delta } \right)h\left( x \right) $$
(1.1)
is contained in the union of finitely many proper linear subspaces of ℙn.
Jan-Hendrik Evertse, Roberto G. Ferretti
On the Diophantine Equation G n (x) = G m (y) with Q (x, y)=0
Abstract
Let K denote an algebraically closed field of characteristic 0, and let A0,..., Ad–1, G0,..., G d- 1 ∈ K[X] and \( \left( {Gn\left( X \right)} \right)_{n = 0}^\infty \) be a sequence of polynomials defined by the d- th order linear recurring relation
$$ G_{n + d} \left( X \right) = A_{d - 1} \left( X \right)G_{n + d - 1} \left( X \right) + \cdots + A_0 \left( X \right)G_n \left( X \right), for n \geqslant 0. $$
(1)
Furthermore, let P(X) ∈ K[X], deg P ≥ 1. Recently, we investigated the question, what can be said about the number of solutions of the Diophantine equation
$$ Gn\left( X \right) = Gm\left( {P\left( X \right)} \right). $$
(2)
Clemens Fuchs, Attila Pethő, Robert F. Tichy
A Criterion for Polynomials to Divide Infinitely Many k- Nomials
Abstract
A polynomial Q ∈ ℚ[x] of the form
$$ Q\left( x \right) = \sum\limits_{i = 1}^k {a_i x^{m_i } } with m_1 > \ldots > m_{k - 1} > m_k = 0 and a_1 = 1 $$
is called a standard k-nomial. It is worth to mention that the restriction to monic k-nomials is only for convenience. We may replace every standard k-nomial by any of its constant multiples, and the theorems would still be valid. We call (m 1 , ..., m k ) the exponent k-tuple of Q. Note that if Q is a standard k-nomial, but not a standard (k-1)-nomial, then its exponent k-tuple is uniquely determined. Let
$$ \begin{gathered} PR_k = \left\{ {P \in \mathbb{Q}\left[ x \right]: \exists Q \in \mathbb{Q}\left[ x \right] and r \in \mathbb{Z} with deg \left( Q \right) < k} \right. \hfill \\ \left. {and r \geqslant 1 such that P\left( x \right)\left| { Q\left( {x^r } \right)over \mathbb{Q}} \right.} \right\}. \hfill \\ \end{gathered} $$
Lajos Hajdu, Robert Tijdeman
Fr. Approximants de Padé des q-Polylogarithmes
Abstrait
Considérons la série
$$ \zeta q\left( s \right) = \sum\limits_{k = 1}^\infty {k^{s - 1} \tfrac{{q^k }} {{1 - q^k }},} $$
qui converge pour tout complexe |q|< 1 et tout entier s ≥ 1. La notation ζq est justifiée par le fait que cette fonction est un q-analogue de la fonction zêta de Riemann ζ (s) au sens suivant (voir [5, paragraphe 4.1], [3, Theorem 2] ou [8]),
$$ \mathop {\lim }\limits_{q \to 1} \left( {1 - q} \right)^s \zeta _q \left( s \right) = \left( {s - 1} \right)!\sum\limits_{k = 1}^\infty {\frac{1} {{k^s }} = \left( {s - 1} \right)!\zeta \left( s \right).} $$
Christian Krattenthaler, Tanguy Rivoal
The Set of Solutions of Some Equation for Linear Recurrence Sequences
Abstract
In [SS1] Schlickewei and Schmidt studied the solutions of various linear equations involving members of recurrence sequences. Most of them are of the form
$$ F_1 \left( {x_1 } \right) + \cdots + F_n \left( {x_n } \right) = 0 $$
(A)
with x i ∈ ℤ, where \( F_j \left( x \right) = \sum\nolimits_{i = 0}^{r_j } {f_{ji} \left( x \right)\alpha _{ji}^x \left( {j = 1, \ldots ,n} \right)} \), r j > 0 with given polynomials f ji and nonzero numbers αji (thus for each j, (F j (x))x∈ℤ is a linear recurrence sequence, see also [ST, Sec.C]). The general assumption of [SS1, p.220] is that αj0 is a root of unity and that f ji ≠ 0 for i > 0 (f j0 may be zero), j = 1, ..., n. Furthermore, they restrict to nondegenerate sequences, i.e., αjijh is not a root of unity for h ≠ i.
Viktor Losert
Counting Algebraic Numbers with Large Height I
Abstract
Let ℚ denote the field of rational numbers, https://static-content.springer.com/image/chp%3A10.1007%2F978-3-211-74280-8_14/978-3-211-74280-8_14_IEq1_HTML.gif an algebraic closure of ℚ, and H : https://static-content.springer.com/image/chp%3A10.1007%2F978-3-211-74280-8_14/978-3-211-74280-8_14_IEq2_HTML.gif the absolute, multiplicative, Weil height. For each positive integer d and real number \( \mathcal{H} \geqslant 1 \), it is well known that the number https://static-content.springer.com/image/chp%3A10.1007%2F978-3-211-74280-8_14/978-3-211-74280-8_14_IEq4_HTML.gif of points α in https://static-content.springer.com/image/chp%3A10.1007%2F978-3-211-74280-8_14/978-3-211-74280-8_14_IEq5_HTML.gif having degree d over ℚ and satisfying \( H\left( \alpha \right) \leqslant \mathcal{H} \) is finite. This is the one-dimensional case of Northcott’s Theorem [8] (see also [5, page 59]). The systematic study of the counting function https://static-content.springer.com/image/chp%3A10.1007%2F978-3-211-74280-8_14/978-3-211-74280-8_14_IEq7_HTML.gif , and that of related functions in higher dimensions, was begun by Schmidt [10]. It is relatively easy to prove the existence of a positive constant C = C(d) such that
https://static-content.springer.com/image/chp%3A10.1007%2F978-3-211-74280-8_14/978-3-211-74280-8_14_Equ1_HTML.gif
(1)
and also the existence of positive constants c = c(d) and \( \mathcal{H}_0 = \mathcal{H}_0 \left( d \right) \) such that
https://static-content.springer.com/image/chp%3A10.1007%2F978-3-211-74280-8_14/978-3-211-74280-8_14_Equ2_HTML.gif
(2)
David Masser, Jeffrey D. Vaaler
Class Number Conditions for the Diagonal Case of the Equation of Nagell and Ljunggren
Abstract
The diagonal case of the Nagell-Ljunggren equation is
$$ \frac{{x^p - 1}} {{x - 1}} = p^e \cdot y^p with x,y \in \mathbb{Z} e \in \left\{ {0,1} \right\}, $$
(1)
and p an odd prime. The only known nontrivial solution is
$$ \frac{{18^3 - 1}} {{18 - 1}} = 7^3 , $$
(2)
and it is conjectured to be also the only such solution. However, it is not even proved that (1) has only finitely many solution.
Preda Mihăilescu
Construction of Approximations to Zeta-Values
Abstract
Polylogarithmic functions are defined by series
$$ L_k \left( z \right) = \sum\limits_{v = 1}^\infty {\frac{{z^v }} {{v^k }}} , k \geqslant 1. $$
Due to equalities Lk;(1) = ζ(k), k ≥ 2, they play an important role in study of arithmetic properties of Riemann zeta-function ζ(s) at integer points.
Yuri V. Nesterenko
Fr. Quelques Aspects Diophantiens des VariéTés Toriques Projectives
Abstrait
Les variétés toriques jouent un rôle important au carrefour de l’algèbre, la géométrie et la combinatoire. Elles constituent une classe de variétés suffisamment rigide pour que beaucoup des invariants s’explicitent en termes combinatoires, et en même temps suffisamment riche pour permettre de tester et illustrer diverses conjectures et théories abstraites. Elle trouve application dans de nombreuses branches des mathématiques : géométrie algébrique bien sûr, algèbre commutative, combinatoire, calcul formel, géométries symplectique et kählerienne, topologie et physique mathématique, voir par exemple [Ful93], [GKZ94], [Stu96], [Cox01], [Aud91], [Don02].
Patrice Philippon, Martín Sombra
Fr. Une Inégalité de Łojasiewicz Arithmétique
Abstrait
Une inégalité de Łojasiewicz minore la valeur |f(x)| d’une fonction analytique f : ℝn ℝ par une puissance de la distance de x à l’ensemble des zéros de f. Nous nous intéressons ici au cas arithmétique où f est un polynôme à coefficients entiers.
Gaël Rémond
On the Continued Fraction Expansion of a Class of Numbers
Abstract
A classical result of Dirichlet asserts that, for each real number ξ and each real X ≥ 1, there exists a pair of integers (x 0 , x1) satisfying
$$ 1 \leqslant x_0 \leqslant X and \left| {x_0 \xi - x_1 } \right| \leqslant X^{ - 1} $$
(a general reference is Chapter I of [10]). If ξ is irrational, then, by letting X tend to infinity, this provides infinitely many rational numbers x 1 /x 0 with |ξ - x1/x0 x 0 - 2. By contrast, an irrational real number ξ is said to be badly approximable if there exists a constant c1 > 0 suchthat |ξ - p/q > c 1 q - 2 for each p/q ∈ or,equivalently,if ξ has bounded partial quotients in its continued fraction expansion. Thanks to H. Davenport and W. M. Schmidt, the badly approximable real numbers can also be described as those ξ ∈ ℝ \ ℚ for which the result of Dirichlet can be improved in the sense that there exists a constant c2 < 1 such that the inequalities 1 ≤ x0X and |x0ξ - x 1 |≤ c2X-1 admit a solution (x0, x1) ∈ ℤ2 for each sufficiently large X (see Theorem 1 of [2]).
Damien Roy
The Number of Solutions of a Linear Homogeneous Congruence
Abstract
The aim of this paper is to propose and to study the following Conjecture. Let n ∈ ℕ, a i ∈ ℤ and b i ∈ ℕ(l ≤ i ≤ k). The number N(n; a 1 , b 1 ; ; a k ,bk) of solutions of the congruence
$$ \sum\limits_{i = 1}^k {a_i x_i \equiv } 0\left( {\bmod n} \right)with 0 \leqslant x_i \leqslant b_i $$
(1)
satisfies the inequality
$$ N\left( {n;a_1 ,b_1 ; \ldots ;a_k ,b_k } \right) \geqslant 2^{1 - n} \prod\limits_{i = 1}^k {\left( {b_i + 1} \right).} $$
(2)
Andrzej Schinzel
A Note on Lyapunov Theory for Brun Algorithm
Abstract
Regular continued fractions exhibit a number of remarkable properties. We mention three of them.
Fritz Schweiger
Orbit Sums and Modular Vector Invariants
Abstract
Let m, n positive integers, R a commutative ring with the unit element 1, and
$$ A_{mn} = R\left[ {x_{11} , \ldots ,x_{m1} ; \ldots ;x_{1n} , \ldots ,x_{mn} } \right] $$
the algebra of polynomials in mn variables xij over R. The symmetric group S n operates on the algebra A mn as a group of R-automorphisms by the rule: σ(xij) = xi,σ(j), σ G. Denote by \( A_{mn}^{S_n } \) the subalgebra of invariants of the algebra A mn with respect to S n and define polarized elementary symmetric polynomials \( u_{r_1 , \ldots ,r_m } \in A_{mn}^{S_n } \) in n vector variables (x11,..., xm1),..., (x1n,..., xmn) by means of the following formal identity
$$ \prod\limits_{j = 1}^n {\left( {1 + x_1 jz_1 + \cdots + x_{mj} z_m } \right) = 1 + \sum\limits_{1 \leqslant r_1 + \cdots + r_m \leqslant n} {u_{r1, \ldots rm} z_1^{r_1 } \cdots z_m^{r_m } .} } $$
Serguei A. Stepanov
New Irrationality Results for Dilogarithms of Rational Numbers
Abstract
A natural method to investigate diophantine properties of transcendental (or conjecturally transcendental) constants occurring in various mathematical contexts consists in the search for sequences of good rational approximations, or algebraic approximations with bounded degree, to suitable values of some special transcendental functions, such as the logarithm, or the polylogarithm of order q ≤ 2, or the hypergeometric functions, etc. Traditionally, one employs for this purpose Padé or Padé-type approximations to the functions involved.
Carlo Viola
Metadaten
Titel
Diophantine Approximation
herausgegeben von
Hans Peter Schlickewei
Klaus Schmidt
Robert F. Tichy
Copyright-Jahr
2008
Verlag
Springer Vienna
Electronic ISBN
978-3-211-74280-8
Print ISBN
978-3-211-74279-2
DOI
https://doi.org/10.1007/978-3-211-74280-8