Recurrence properties of sequences of integers
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.
scientific article; zbMATH DE number 5712106
scientific article