Abstract
Aiming at a simultaneous extension of Khintchine’s and Furstenberg’s Recurrence theorems, we address the question if for a measure preserving system \((X,\mathcal{X},\mu,T)\) and a set \(A\in\mathcal{X}\) of positive measure, the set of integers n such that \(\mu(A{\cap} T^{n}A{\cap} T^{2n}A{\cap} \ldots{\cap} T^{kn}A)>\mu(A)^{k+1}-\epsilon\) is syndetic. The size of this set, surprisingly enough, depends on the length (k+1) of the arithmetic progression under consideration. In an ergodic system, for k=2 and k=3, this set is syndetic, while for k≥4 it is not. The main tool is a decomposition result for the multicorrelation sequence \(\int{f(x)f(T^{n}x)f(T^{2n}x){\ldots} f(T^{kn}x) \,d\mu(x)}\), where k and n are positive integers and f is a bounded measurable function. We also derive combinatorial consequences of these results, for example showing that for a set of integers E with upper Banach density d*(E)>0 and for all ε>0, the set
is syndetic.
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Bergelson, V., Host, B., Kra, B. et al. Multiple recurrence and nilsequences. Invent. math. 160, 261–303 (2005). https://doi.org/10.1007/s00222-004-0428-6
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DOI: https://doi.org/10.1007/s00222-004-0428-6