  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.
