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We investigate how spectral properties of a measure preserving system$(X,\mathcal{B},\mu,T)$ are reflected in the multiple ergodic averages arisingfrom that system. For certain sequences $a:\mathbb{N}\to\mathbb{N}$ we providenatural 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. Inparticular, our results apply to infinite arithmetic progressions $a(n)=qn+r$,Beatty sequences $a(n)=\lfloor \theta n+\gamma\rfloor$, the sequence ofsquarefree numbers $a(n)=q_n$, and the sequence of prime numbers $a(n)=p_n$. Wealso obtain a new refinement of Szemer\'edi's theorem via Furstenberg'scorrespondence principle.  ERRATUM: Theorem 7.1 in the paper is incorrect as stated, and the errororiginates with Proposition 7.5, part (iii), which was incorrectly quoted fromthe literature. In the attached erratum we fix the problem by establishing aslightly weaker version of Theorem 7.1 and use it to give a new proof ofTheorem 4.2. This ensures that all main results in our main article remaincorrect. We thank Zhengxing Lian and Jiahao Qiu for bringing this mistake toour attention.

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