Ruzsa's problem on sets of recurrence
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.
existence of Poincaré sequences of integers which are not van der Corput sets
scientific article; zbMATH DE number 4047835