Multiple recurrence and nilsequences

Aiming at a simultaneous extension of Khintchine's and Furstenberg's Recurrence theorems, we address the question if for a measure preserving system (X,χμT) and a set A ∈ χ of positive measure, the set of integers n such that μ (A ∩ Tⁿ A ∩ T²ⁿ ... ∪ T ^kn A) > μ (A)^k+1 - ∈ 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 ∫ f(x)f(Tⁿx) f(T ²ⁿx) ... f(T^knx) dμ (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 ∈ ;> 0, the set {n ∈ ℤ: d* (E ∪ ( E + n) ∪ (E + 2n) ∪ (E + 3n)) > d* (E)⁴ - ∈} is syndetic.