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}
}
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… 
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