Distribution modulo 1 of some oscillating sequences
type: Item
instance of: scholarly article
title: Distribution modulo 1 of some oscillating sequences
zbMATH Open document ID: 0726.11042
DOI: 10.1007/BF02811882
author: Daniel Berend
author: G. A. Kolesnik
published in: Israel Journal of Mathematics
publication date: 1990-01-01T00:00:00Z
review text: The distribution mod 1 of sequences of the form P(n)F(Q(n)), \(n=1,2,..\). is studied, where P and Q are polynomials and F is a sufficiently many times differentiable periodic function. For example Theorem 2.2 states that for any given nonnegative integers d and e, there exist numbers \(s=s(d,e)\) and \(r=r(d,e)>0\) possessing the following property: For any polynomials P and Q of degrees d and e, respectively, and any periodic function F which is s times differentiable at Q(0) the inequality \[ \| P(n)F(Q(n))-P(0)F(Q(0))\| <n^{-r} \] has infinitely many positive integer solutions. When Q is restricted being a monomial then a minor extra condition on F provides that the values P(n)F(Q(n)) are dense mod 1 while for linear Q and smooth F those values are uniformly distributed mod 1.
Mathematics Subject Classification ID: 11J71
Mathematics Subject Classification ID: 11B83
zbMATH DE Number: 4198101
zbMATH Keywords: uniform distribution
zbMATH Keywords: exponential sums
reviewed by: Antal Balog
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02811882
OpenAlex ID: W4250324063
description: scientific article; zbMATH DE number 4198101