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arXiv · 2609.00405

Multiple ergodic averages for commuting multiplicative actions

Dimitrios Charamaras·Andreas Koutsogiannis·Konstantinos Tsinas
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Abstract

We study the convergence of multiple ergodic averages involving several commuting multiplicative actions. We prove that for finitely generated systems, the averages $$\frac{1}{N}\sum_{n=1}^N S_{1,n}F_1\cdot\ldots\cdot S_{\ell,n}F_\ell$$ converge in norm, settling a conjecture of Frantzikinakis. Our methods rely on a novel generalization of K\'atai'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.

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Dimitrios Charamaras, Andreas Koutsogiannis, Konstantinos Tsinas. 2026-08-31. Multiple ergodic averages for commuting multiplicative actions. https://arxiv.org/abs/2609.00405

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