Multiple recurrence and nilsequences

Aiming at a simultaneous extension of Khintchine(X,X,m,T)(X,mathcal{X},mu,T) and a set A ? XAinmathcal{X} of positive measure, the set of integers n such that A T^2nA T^knA)(A)^k+1-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 kòf(x)f(Tⁿx)f(T²ⁿx)? f(T^knx)  dm(x)int{f(x)f(T^{n}x)f(T^{2n}x){ldots} f(T^{kn}x) ,dmu(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 {n ? mathbbZcolon d^*(E?(E+n)?(E+2n)?(E+3n)) > d^*(E)⁴-e}big{ninmathbb{Z}{colon} d^*big(Ecap(E+n)cap(E+2n)cap(E+3n)big) > d^*(E)^4-epsilonbig}