Metrical theorems on fractional parts on real sequences. II
type: Item
instance of: scholarly article
title: Metrical theorems on fractional parts on real sequences. II
zbMATH Open document ID: 0781.11029
author: Glyn Harman
published in: Journal of Number Theory
publication date: 1994-02-23T00:00:00Z
review text: The author continues his study [J. Reine Angew. Math. 396, 192-211 (1989; Zbl 0657.10055)] on fractional parts of real sequences. Let \(a_ n\) be a sequence of reals satisfying (a) \(a_{n+1}-a_ n\geq\lambda\) (or \(\leq -\lambda\)) for some \(\lambda>0\) and for all sufficiently large \(n\); (b) \(| a_ n|>cn^{1+\delta}\) for some \(c,\delta>0\); (c) \(a_ m/a_ n<Km^{-1} n^{-\mu}\) for \(m\leq n^ \sigma\) and for some \(K,\mu,\sigma>0\). The author proves that for almost all real \(x\) there are infinitely many solutions to \[ | a_ n x-m|<(n\log\log(9n))^{-1}\tag{1} \] where \(m\) has no prime factor smaller than \(n^{1+\varepsilon}\) with \(\varepsilon=\min\{\sigma,\delta, \mu\}/4\). Moreover, if \(| a_ n|<n^ A\) for some fixed \(A\), then there are infinitely many solutions to (1) with \(m\) as a prime. The author points out that there are only finitely many solutions for almost all real \(x\) to \[ | a_ n x-m|<n^{-1}(\log\log(9n))^{-1-\eta} \] for any \(\eta>0\) with the above restriction on \(m\). Therefore the above result is very near to best possible. Many useful corollaries can be deduced from the above result. For example, (a) For almost all real \(x\) there are infinitely many solutions to \[ \| a_ n x\|< (n\log\log(9n))^{-1} \] where \(\|\alpha\|=\min\{| \alpha-n|\): \(n\in\mathbb{Z}\}\) for real \(\alpha\). (b) For almost all real \(x\) there are infinitely many fractions \(p/q\) with \[ | x-(p/q)|< \log(q) (q^{3/2}\log\log(q))^{-1} \] where \(p\) is a prime and \(q\) is the square of a prime.
Mathematics Subject Classification ID: 11J54
Mathematics Subject Classification ID: 11J83
Mathematics Subject Classification ID: 11J71
Mathematics Subject Classification ID: 11K60
zbMATH DE Number: 205455
zbMATH Keywords: small fractional parts
reviewed by: Ming-Chit Liu
MaRDI profile type: Publication
full work available at URL: jnth.1993.1033
OpenAlex ID: W1996254210
DOI: 10.1006/JNTH.1993.1033
description: scientific article; zbMATH DE number 205455