# Multiple recurrence and nilsequences

Abstract : Aiming at a simultaneous extension of Khintchine's and Furstenberg's Recurrence theorems, we address the question if for a measure preserving system $(X,\CX,\mu,T)$ and a set $A\in\CX$ of positive measure, the set of integers $n$ such that $\mu(A\cap T^nA\cap T^{2n}A\cap \ldots\cap T^{kn}A) >\mu(A)^{k+1}-\epsilon$ 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 $k\geq 4$ it is not. The main tool is a decomposition result for the multicorrelation sequence $\int f(x)f(T^nx)f(T^{2n}x)\ldots f(T^{kn}x) \,d\mu(x)$, 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 $\epsilon > 0$, the set $\{n\in\Z\colon d^*\bigl(E\cap (E+n)\cap (E+2n)\cap (E+3n)\bigr)> d^*(E)^4-\epsilon\}$ is syndetic.
Document type :
Journal articles

https://hal-upec-upem.archives-ouvertes.fr/hal-01267076
Contributor : Bernard Host <>
Submitted on : Wednesday, February 3, 2016 - 7:45:00 PM
Last modification on : Thursday, March 19, 2020 - 12:26:02 PM

### Identifiers

• HAL Id : hal-01267076, version 1

### Citation

Vitaly Bergelson, Bernard Host, Bryna Kra. Multiple recurrence and nilsequences. Inventiones Mathematicae, Springer Verlag, 2005, 160, pp.261-303. ⟨hal-01267076⟩

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