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

Abstract

We investigate how spectral properties of a measure-preserving system$(X,{\mathcal{B}},\unicode[STIX]{x1D707},T)$are reflected in the multiple ergodic averages arising from that system. For certain sequences$a:\mathbb{N}\rightarrow \mathbb{N}$, we provide natural conditions on the spectrum$\unicode[STIX]{x1D70E}(T)$such that, for all$f_{1},\ldots ,f_{k}\in L^{\infty }$,$$\begin{eqnarray}\lim _{N\rightarrow \infty }\frac{1}{N}\mathop{\sum }_{n=1}^{N}\mathop{\prod }_{j=1}^{k}T^{ja(n)}f_{j}=\lim _{N\rightarrow \infty }\frac{1}{N}\mathop{\sum }_{n=1}^{N}\mathop{\prod }_{j=1}^{k}T^{jn}f_{j}\end{eqnarray}$$in$L^{2}$-norm. In particular, our results apply to infinite arithmetic progressions,$a(n)=qn+r$, Beatty sequences,$a(n)=\lfloor \unicode[STIX]{x1D703}n+\unicode[STIX]{x1D6FE}\rfloor$, the sequence of squarefree numbers,$a(n)=q_{n}$, and the sequence of prime numbers,$a(n)=p_{n}$. We also obtain a new refinement of Szemerédi’s theorem via Furstenberg’s correspondence principle.