Inhomogeneous diophantine approximations and distribution of fractional parts
type: Item
instance of: scholarly article
title: Inhomogeneous diophantine approximations and distribution of fractional parts
zbMATH Open document ID: 1296.11094
author: Anton Shutov
published in: Journal of Mathematical Sciences (New York)
publication date: 2012-11-28T00:00:00Z
review text: Let \(\alpha\) be an irrational number with continued fraction expansion \(\alpha=[0;q_1,q_2,\dots]\), \(I\) be an interval in \([0,1)\) with the length \(|I|\), \(\{n\alpha\}\) be a fractional part of \(n\alpha\), \(n=1,2,\dots\), and denote the local discrepancy function \(r(\alpha,a,n,I)=|\#\{1\leq k\leq n;\{k\alpha + a\}\in I\}-n|I||\). Let \(|I|=\beta\). In the present paper the author obtain a bound for \(r(\alpha,a,n,I)\), considering the sequence of best \(\alpha\)-approximations to \(\beta\) as follows: \[ A_{\alpha\beta}(0)=\begin{cases} 0&\text{if }\beta\leq\frac{1}{2}\\ 1&\text{if }\beta>\frac{1}{2}\end{cases}; \] \[ A_{\alpha\beta}(m+1)=\min\{k\in\mathbb N;\|k\alpha-\beta\|<\|A_{\alpha\beta}(m)\alpha-\beta\|\}; \] \[ a_{\alpha\beta}(n)=\min\{m;A_{\alpha\beta}(m)>n\}; \] \[ \tilde a(n)=A_{\alpha\beta}(a_{\alpha\beta}(n)); \] \[ \psi(n)=\bigg[\log_{\frac{1+\sqrt{5}+1}{2}}\bigg(\sqrt{5}n+\sqrt{5}+\frac{1}{2}\bigg)\bigg]+1. \] The main result of the present paper is \[ r(\alpha,a,n,I)\leq\frac{1}{2}\bigg(\max_{1\leq j\leq\psi(\tilde a(n))+2}q_j+15\bigg)(a_{\alpha\beta}(n)+1). \] In the proof the author uses generalized Fibonacci tilings as in his previous papers [in: Analytic and probabilistic methods in number theory. Proceedings of the 4th international conference in honour of J. Kubilius, Palanga, Lithuania, September 25--29, 2006. Vilnius: TEV. 190--203 (2007; Zbl 1165.11062)] and [Chebyshevskiĭ Sb. 7, No. 3(19), 110--128 (2006; Zbl 1241.11091)].
reviewed by: Oto Strauch
Mathematics Subject Classification ID: 11K38
Mathematics Subject Classification ID: 11K31
zbMATH DE Number: 6110759
zbMATH Keywords: local discrepancy
zbMATH Keywords: continued fraction
zbMATH Keywords: fractional parts of \(n\alpha\) sequence
zbMATH Keywords: Hecke-Kesten problem
MaRDI profile type: Publication
full work available at URL: s10958-012-0762-y
OpenAlex ID: W2009952285
cites work: Répartition des suites (nα)n∈Net substitutions
cites work: Sequences, discrepancies and applications
cites work: On a conjecture of Erdös and Szüsz related to uniform distribution mod 1
cites work: Q4767350
cites work: On sums of fractional parts \(\{na+y\}\)
cites work: Q5695330
cites work: Q5504811
cites work: Q5564439
DOI: 10.1007/S10958-012-0762-Y
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description: scientific article; zbMATH DE number 6110759