Recurrence properties of sequences of integers
type: Item
instance of: scholarly article
title: Recurrence properties of sequences of integers
zbMATH Open document ID: 1193.28013
author: Dominique Schneider
author: Ai-Hua Fan
published in: Science China. Mathematics
publication date: 2010-05-26T00:00:00Z
review text: A (strictly increasing) sequence \((u_n)\) of natural numbers is called recurrent if, for any probability measure-preserving system \((Y,\mathcal{B},\nu,S)\) and set \(A\) of positive \(\nu\)-measure there is an \(n\) with \(\nu(A\cap S^{-u_n}A)>0\), is called \(L^2\)-good if the ergodic average \((1/N)\sum_{n=1}^{N}g(S^{u_n})\) converge in \(L^2(\nu)\) for any \(G\in L^2(\nu)\), and is called \(L^2\)-exact if the limit is the conditional expectation of \(g\) with respect to the invariant \(\sigma\)-algebra of \(S\). The latter two properties are characterised in terms of properties of the Fourier transform \((1/N)\sum_{n=1}^{N}\exp(2\pi iu_nt)\). Several relationships are shown here: \(L^2\)-exact sequences are recurrent, the sequence of return times of a point \(x\) into a set in a weak-mixing system is \(L^2\)-exact for almost every \(x\) (and if, in addition, the system is uniquely ergodic, then for every \(x\)). Results are also found for the relationship with \(\Theta\)-Hartman sequences and some random sequences. The methods used include Fourier analysis, Wiener--Wintner, spectral methods, and versions of the van der Corput lemma.
Mathematics Subject Classification ID: 28D05
Mathematics Subject Classification ID: 37A45
Mathematics Subject Classification ID: 60G50
zbMATH DE Number: 5712106
zbMATH Keywords: recurrence
zbMATH Keywords: ergodic theorems
zbMATH Keywords: Hartman sequence
zbMATH Keywords: random sequence
zbMATH Keywords: Wiener--Wintner theorem
reviewed by: Thomas B. Ward
MaRDI profile type: Publication
full work available at URL: s11425-010-0044-4
OpenAlex ID: W3145658391
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DOI: 10.1007/S11425-010-0044-4
description: scientific article; zbMATH DE number 5712106