Ruzsa's problem on sets of recurrence
type: Item
instance of: scholarly article
title: Ruzsa's problem on sets of recurrence
zbMATH Open document ID: 0643.10045
DOI: 10.1007/BF02787258
published in: Israel Journal of Mathematics
publication date: 1987-01-01T00:00:00Z
review text: A set H of integers is called intersective, if \(H\cap (S-S)\neq \emptyset\), for every set S of positive density. H is correlative, if for every sequence \((y_ n)\) of complex numbers satisfying \(\sum_{n\leq x}| y_ n|^ 2=O(x)\) and \(\sum_{n\leq x}\bar y_ ny_{n+h}=o(x)\) for all \(h\in H\), \(\sum_{n\leq x}y_ n=o(x)\) must hold. Finally, H is approximative, if for every finite collection \((u_ 1,...,u_ n)\) of real numbers and \(\epsilon >0\) there is an \(h\in H\) such that \(\| u_ jh\| <\epsilon\) for \(j=1,...,n\), where \(\| \|\) is the distance from the nearest integer. The implications correlative \(\to inter\sec tive \to approximative\) are known to hold. Intersectivity is equivalent to being a Poincaré or recurrent set; correlativity is equivalent to van der Corput's property and other equivalent forms are also known [see \textit{T. Kamae} and \textit{M. Mendès France}, Isr. J. Math. 31, 335-342 (1978; Zbl 0396.10040) and the reviewer, Colloq. Math. Soc. Janos Bolyai 34, 1419-1443 (1984; Zbl 0572.10035)]. The main result of this paper is that correlativity and intersectivity are \textit{not} equivalent. The related fact that intersectivity and approximativity are not equivalent either was recently proved by \textit{I. Kriz} [Graphs and Comb. 3, 145-158 (1987)]. The proof is based on a sophisticated and ingenious construction. It is also shown that for random (or ``generic'') sets these properties are equivalent.
Mathematics Subject Classification ID: 11B13
Mathematics Subject Classification ID: 05B10
Mathematics Subject Classification ID: 28D99
Mathematics Subject Classification ID: 11K99
Mathematics Subject Classification ID: 47A35
zbMATH DE Number: 4047835
zbMATH Keywords: existence of Poincaré sequences of integers which are not van der Corput sets
zbMATH Keywords: recurrent set
zbMATH Keywords: Poincaré set
zbMATH Keywords: difference sets
zbMATH Keywords: correlative sets
zbMATH Keywords: correlativity
zbMATH Keywords: intersectivity
zbMATH Keywords: approximativity
author: Jean Bourgain
reviewed by: Imre Z. Ruzsa
MaRDI profile type: Publication
cites work: Ensembles intersectifs et récurrence de Poincaré. (Intersective sets and Poincaré recurrence)
cites work: Q3909268
cites work: Van der Corput's difference theorem
cites work: Q5182631
cites work: Q4121006
cites work: Q3856841
full work available at URL: https://doi.org/10.1007/bf02787258
OpenAlex ID: W1974504121
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description: scientific article; zbMATH DE number 4047835