A spectral refinement of the Bergelson–Host–Kra decomposition and new multiple ergodic theorems

We investigate how spectral properties of a measure-preserving system (X, B, mu, T) are reflected in the multiple ergodic averages arising from that system. For certain sequences a :N -> N, we provide natural conditions on the spectrum sigma (T) such that, for all f(1), ..., f(k) is an element of L-infinity, lim(N ->infinity) 1/N Sigma(N)(n=1) Pi(k)(j=1) T-ja(n) f(j) = lim(N ->infinity) 1/N Sigma(N)(n=1) Pi(k)(j=1) T-jn fj in L-2-norm. In particular, our results apply to infinite arithmetic progressions, a(n) = qn + r, Beatty sequences, a(n) = [theta n + gamma], the sequence of squarefree numbers, a(n) = qn , and the sequence of prime numbers, a(n) = p(n). We also obtain a new refinement of Szemeredi's theorem via Furstenberg's correspondence principle.