On irregularities of distribution in shifts and dilations of integer sequences. I
type: Item
instance of: scholarly article
title: On irregularities of distribution in shifts and dilations of integer sequences. I
zbMATH Open document ID: 0591.10045
DOI: 10.1007/BF01450834
published in: Mathematische Annalen
publication date: 1987-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164204
review text: Authors' review: The goal of this series is to study, for any sequence of complex numbers \(\varepsilon_ 1,...,\varepsilon_ N\) and any sequence of positive integers \(b_ 1,b_ 2,...\), the quantity \(\max | \sum^{t}_{j=1}\epsilon_{a+b_ jq}|\) where the maximum is taken over all integers \(a\) and all positive integers \(q\) and \(t\). In the above sum we take only those terms \(\varepsilon_{a+b_ jq}\) for which \(1\leq a+b_ jq\leq N\). For example, let \(N, L\) and \(b_ 1,...,b_ t\) be positive integers with \(b_ 1<b_ 2<...<b_ t\leq L\) and \(L\leq N^{1/2}\). Let \(\varepsilon_ 1,...,\varepsilon_ N\) be complex numbers of absolute value at least one. We prove that \[ \max_{1\leq q\leq 14L;\quad -14L^ 2\leq m\leq N}| \sum^{t}_{j=1}\epsilon_{m+b_ jq}| \geq t/(30 L^{1/2}). \] Further, let \(\delta\) be a positive real number, and let \(\varepsilon_ 1,...,\varepsilon_ N\) be complex numbers of absolute value at least one. We show that there exists a number \(c(\delta)\) which is effectively computable in terms of \(\delta\) such that if \(N>c(\delta)\) then \[ \max_{1\leq q,t\leq N^{1/3};\quad -N\leq m\leq N}| \sum^{t}_{x=1}\epsilon_{m+x^ 2q}| >N^{1/6}\quad \exp (- (1+\delta)(\log 2\log N)/3\log \log N). \] We apply results of the above type to give lower bounds for character sums.
Mathematics Subject Classification ID: 11K38
Mathematics Subject Classification ID: 11L40
zbMATH DE Number: 3948404
zbMATH Keywords: irregularities of distribution
zbMATH Keywords: shifts
zbMATH Keywords: dilations
zbMATH Keywords: integer sequences
zbMATH Keywords: sequence of complex numbers
zbMATH Keywords: lower bounds for character sums
Wikidata QID: Q105956235
author: András Sárközy
author: C. L. Stewart
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3948404