Degree lowering for ergodic averages along arithmetic progressions

We examine the limiting behavior of multiple ergodic averages associated with arithmetic progressions whose differences are elements of a fixed integer sequence. For each ℓ, we give necessary and sufficient conditions under which averages of length ℓ of the aforementioned form have the same limit as averages of ℓ-term arithmetic progressions. As a corollary, we derive a sufficient condition for the presence of arithmetic progressions with length ℓ+1 and restricted differences in dense subsets of integers. These results are a consequence of the following general theorem: in order to verify that a multiple ergodic average is controlled by the degree d Gowers–Host–Kra seminorm, it suffices to show that it is controlled by some Gowers–Host–Kra seminorm, and that the degree d control follows whenever we have degree d + 1 control. The proof relies on an elementary inverse theorem for the Gowers–Host–Kra seminorms involving dual functions, combined with novel estimates on averages of seminorms of dual functions. We use these estimates to obtain a higher order variant of the degree lowering argument previously used to cover averages that converge to the product of integrals.