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

We investigate how spectral properties of a measure-preserving system $(X,{\mathcal{B}},\unicode[STIX]{x1D707},T)$ 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.