Ergodic theorems for nonconventional arrays and an extension of the szemerédi theorem

The paper is primarily concerned with the asymptotic behavior as N → ∞ of averages of nonconventional arrays having the form N−^{1 N}_n=1_j=1 T^Pj(n,N⁾f_j where f_j’s are bounded measurable functions, T is an invertible measure preserving transformation and P_j’s are polynomials of n and N taking on integer values on integers. It turns out that when T is weakly mixing and P_j(n, N) = p_jn + q_jN are linear or, more generally, have the form P_j(n, N) = P_j(n) + Q_j(N) for some integer valued polynomials P_j and Q_j then the above averages converge in L² but for general polynomials P_j of both n and N the L² convergence can be ensured even in the “conventional” case = 1 only when T is strongly mixing while for > 1 strong 2-mixing should be assumed. Studying also weakly mixing and compact extensions and relying on Furstenberg’s structure theorem we derive an extension of Szemerédi’s theorem saying that for any subset of integers Λ with positive upper density there exists a subset N_Λ of positive integers having uniformly bounded gaps such that for N ∈ N_Λ and at least εN, ε > 0 of n’s all numbers p_jn + q_jN, j = 1, ..., , belong to Λ. We obtain also a version of these results for several commuting transformations which yields a corresponding extension of the multidimensional Szemerédi theorem.