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We examine the limiting behavior of multiple ergodic averages associated witharithmetic progressions whose differences are elements of a fixed integersequence. For each $\ell$, we give necessary and sufficient conditions underwhich averages of length $\ell$ of the aforementioned form have the same limitas averages of $\ell$-term arithmetic progressions. As a corollary, we derive asufficient condition for the presence of arithmetic progressions with length$\ell+1$ and restricted differences in dense subsets of integers. These resultsare a consequence of the following general theorem: in order to verify that amultiple ergodic average is controlled by the degree $d$ Gowers-Host-Kraseminorm, it suffices to show that it is controlled by some Gowers-Host-Kraseminorm, and that the degree $d$ control follows whenever we have degree $d+1$control. The proof relies on an elementary inverse theorem for theGowers-Host-Kra seminorms involving dual functions, combined with novelestimates on averages of seminorms of dual functions. We use these estimates toobtain a higher order variant of the degree lowering argument previously usedto cover averages that converge to the product of integrals.

Degree lowering for ergodic averages along arithmetic progressions