• DOI:10.1007/S00222-004-0428-6
• Corpus ID: 1380361

Multiple recurrence and nilsequences

@article{Bergelson2005MultipleRA,
  title={Multiple recurrence and nilsequences},
  author={Vitaly Bergelson and Bernard Host and Bryna Kra and Imre Z. Ruzsa},
  journal={Inventiones mathematicae},
  year={2005},
  volume={160},
  pages={261-303}
}

• V. Bergelson, B. Host, +1 author I. Ruzsa
• Published 1 March 2005
• Mathematics
• Inventiones mathematicae

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 $$\big\{n\in\mathbb{Z}{\colon} d^*\big(E\cap(E+n)\cap(E+2n)\cap(E+3n)\big) > d^*(E)^4-\epsilon\big\}$$ is syndetic.