Multiple ergodic averages for commuting multiplicative actions

Abstract

We study the convergence of multiple ergodic averages involving several commuting multiplicative actions. We prove that for finitely generated systems, the averages $$\frac{1}{N}\sum _{n=1}^N{S}_{1,n}{F}_1\cdot \dots \cdot S_{\ell ,n}{F}_{\ell }$$ converge in norm, settling a conjecture of Frantzikinakis. Our methods rely on a novel generalization of Kátai's orthogonality criterion that allows us to obtain box seminorm control, the machinery of magic extensions originating in the work of Host, and a delicate induction on the complexity of the initial averages that ultimately reduces our problem to well-known mean convergence results.

35 pages