On random perturbation of some intersective sets.
type: Item
instance of: scholarly article
title: On random perturbation of some intersective sets.
zbMATH Open document ID: 1065.37004
DOI: 10.1016/S0019-3577(04)80006-5
author: Yanyan Li
published in: Indagationes Mathematicae. New Series
publication date: 2005-01-03T00:00:00Z
review text: Let \(S\) be a subset of \(\mathbb{Z}\) and let \(I= (I_n)_{n\geq 1}\) be a sequence of intervals in \(\mathbb{Z}\) whose lengths are increasing. Set \[ b(S, I)= \limsup_{n\to\infty}\, {|S\cap I_n|\over|I_n|}. \] The Banach density of \(S\), denoted by \(b(S)\), is defined by \(b(S)= \sup_Ib(S,I)\), where the supremum is taken over all sequences \(I\) of intervals in \(\mathbb{Z}\) whose lengths increase. A sequence of nonnegative integers \({\mathbf k}= (k_n)_{n\geq 1}\) is said to be multiply intersective if given any subset \(E\) of \(\mathbb{N}\) with positive Banach density \(b(E)\), there exists a subset \(R\) of \(\mathbb{N}\) with \[ \lim_{N\to\infty}\, {|R\cap[1, N]|\over N}\geq b(E), \] such that for any subset \(\{n_1,\dots, n_r\}\) of \(R\) we have \[ b(E\cap (E+ k_{n_1})\cap\cdots\cap(E+ k_{n_r}))> 0. \] The authos prove the following result: If \({\mathbf k}= (k_n)_{n\geq 1}\) is a multiply intersective set, then this is also the case of \((k_n+ \theta_n)_{n\geq 1}\), for almost all \(\mathbb{N}\)-valued independent identically distributed random variables \(\theta= (\theta_n)_{n\geq 1}\).
Mathematics Subject Classification ID: 37A25
Mathematics Subject Classification ID: 11K99
Mathematics Subject Classification ID: 28D05
Mathematics Subject Classification ID: 37A30
Mathematics Subject Classification ID: 37A45
Mathematics Subject Classification ID: 11K50
zbMATH DE Number: 2124777
zbMATH Keywords: multiply intersection sets
zbMATH Keywords: Poincaré recurrence
zbMATH Keywords: Banach density
zbMATH Keywords: ergodic theorem
reviewed by: Boris Adamczewski
MaRDI profile type: Publication
cites work: Sets of Recurrence of Zm -Actions and Properties of Sets of Differences in Zm
cites work: Ensembles intersectifs et récurrence de Poincaré. (Intersective sets and Poincaré recurrence)
cites work: Ruzsa's problem on sets of recurrence
cites work: Q3909268
cites work: Q5800881
cites work: Van der Corput's difference theorem
cites work: Subadditive mean ergodic theorems
cites work: On uniformly distributed sequences of integers and Poincaré recurrence
cites work: On uniformly distributed sequences of integers and Poincaré recurrence. II
cites work: On certain sets of integers and intersectivity.
cites work: A Combinatorial Application of the Maximal Ergodic Theorem
cites work: Q3856841
cites work: On difference sets of sequences of integers. I
cites work: Estimating random polynomials by means of metric entropy methods
full work available at URL: s0019-3577(04)80006-5
OpenAlex ID: W2009874254
description: scientific article; zbMATH DE number 2124777