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 $(X,\mathcal{B},\mu,T)$ are reflected in the multiple ergodic averages arising from that system. For certain sequences $a:\mathbb{N}\to\mathbb{N}$ we provide natural conditions on the spectrum $\sigma(T)$ such that for all $f_1,\ldots,f_k\in L^\infty$, \begin{equation*} \lim_{N\to\infty} \frac{1}{N} \sum_{n=1}^N \prod_{j=1}^k T^{ja(n)}f_j = \lim_{N\rightarrow\infty} \frac{1}{N} \sum_{n=1}^N \prod_{j=1}^k T^{jn}f_j \end{equation*} in $L^2$-norm. In particular, our results apply to infinite arithmetic progressions $a(n)=qn+r$, Beatty sequences $a(n)=\lfloor \theta n+\gamma\rfloor$, 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 Szemer\'edi's theorem via Furstenberg's correspondence principle. ERRATUM: Theorem 7.1 in the paper is incorrect as stated, and the error originates with Proposition 7.5, part (iii), which was incorrectly quoted from the literature. In the attached erratum we fix the problem by establishing a slightly weaker version of Theorem 7.1 and use it to give a new proof of Theorem 4.2. This ensures that all main results in our main article remain correct. We thank Zhengxing Lian and Jiahao Qiu for bringing this mistake to our attention.