Small values of \(n^ 2\alpha\pmod 1\)
type: Item
instance of: scholarly article
title: Small values of \(n^ 2\alpha\pmod 1\)
zbMATH Open document ID: 0827.11040
DOI: 10.1007/BF01884304
author: Alexandru Zaharescu
published in: Inventiones Mathematicae
publication date: 1995-11-20T00:00:00Z
full work available at URL: https://eudml.org/doc/144304
review text: We prove that for any real number \(\alpha\), any \(\varepsilon >0\) and any \(N\geq 1\) one has: \[ \min_{1\leq n\leq N} |n^2 \alpha|\;\ll_\varepsilon\;N^{-(4/7)+ \varepsilon}. \] Moreover, \(|n^2 \alpha|< n^{-(2/3)+ \varepsilon}\) for infinitely many positive integers \(n\). Here \(|\cdot |\) stands for the distance to the nearest integer and the implied constant in the first inequality depends only on \(\varepsilon\).
Mathematics Subject Classification ID: 11J71
Mathematics Subject Classification ID: 11J25
zbMATH DE Number: 807309
zbMATH Keywords: distribution of sequences
zbMATH Keywords: fractional parts
zbMATH Keywords: uniform distribution
MaRDI profile type: Publication
cites work: The least quadratic non residue
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OpenAlex ID: W2996745179
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description: scientific article; zbMATH DE number 807309