Probabilistic diophantine approximation. I: Kronecker sequences
type: Item
instance of: scholarly article
title: Probabilistic diophantine approximation. I: Kronecker sequences
zbMATH Open document ID: 0820.11045
DOI: 10.2307/2118607
published in: Annals of Mathematics. Second Series
publication date: 1995-09-14T00:00:00Z
review text: [The first printing of this article in Ann. Math., II. Ser. 140, No. 1, 109-160 (1994) has been with drawn by the editors.] Let \(\alpha\) be a real number and denote by \[ \Delta (\alpha, N)= \max_{1\leq n\leq N} \biggl| \sum_{\substack{ 1\leq i\leq n\\ 0\leq i\alpha< x\bmod 1}} 1- nx \biggr| \] the discrepancy function of the sequence \((n\alpha)\) modulo 1. The investigation of this function is of central interest in diophantine approximation and in the theory of uniform distribution modulo 1. Khintchine (1923) proved that for almost all \(\alpha\) the discrepancy function \(\Delta (\alpha, N)\) is \(O(\log N(\log \log N)^{1+ \varepsilon})\) for arbitrary \(\varepsilon>0\). \textit{W. Schmidt} [Trans. Am. Math. Soc. 110, 493-518 (1964; Zbl 0199.094)] could extend this result of Khintchine to higher dimensions. He proved for the discrepancy function \(\Delta (\alpha, N)\) of the point sequence \((\alpha n)\) in \(\mathbb{R}^ k\bmod 1\), \(\alpha= (\alpha_ 1, \dots, \alpha_ k)\), the metrical result: \[ \Delta (\alpha, N)= O\bigl( (\log N)^{k+1+ \varepsilon} \bigr) \quad \text{for almost all} \quad \alpha\in \mathbb{R}^ k. \] This is the best possible bound via the Erdős- Turan-Koksma inequality. In the present paper the author makes use of a completely different method to obtain the following criterion: \[ \Delta (\alpha, N)= O\bigl( (\log N)^ k \varphi (\log\log N)\bigr) \iff \sum_{n=1}^ \infty {1\over {\varphi (n)}} <\infty, \] where \(\varphi\) is an arbitrary positive increasing function. Setting \(\varphi (x)= x(\log x)^{1+ \varepsilon}\) yields an improvement of Schmidt's bound by the ``critical'' factor \(\log N\), which comes from the Erdős-Turan- Koksma inequality. The proof of this very important result, which reaches the conjectured best possible order of the discrepancy function (apart from log log- factors) is based on a Fourier analysis approach. The Poisson summation formula yields an explicit representation of the ``local'' discrepancy function. Fr…
Mathematics Subject Classification ID: 11K36
Mathematics Subject Classification ID: 11K06
zbMATH DE Number: 742789
zbMATH Keywords: discrepancy function
zbMATH Keywords: uniform distribution
zbMATH Keywords: improvement of Schmidt's bound
zbMATH Keywords: best possible order
zbMATH Keywords: Fourier analysis
zbMATH Keywords: Poisson summation formula
author: József Beck
reviewed by: Robert F. Tichy
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2307/2118607
OpenAlex ID: W4234316157
description: scientific article; zbMATH DE number 742789