PreprintCC BY 4.0arXiv
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 1/N∑n=1N S1,nF1·…· Sℓ,nFℓ 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.
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BibTeX
@misc{charamaras2026multiple,
title = {{Multiple ergodic averages for commuting multiplicative actions}},
author = {Dimitrios Charamaras and Andreas Koutsogiannis and Konstantinos Tsinas},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.00405},
eprint = {2609.00405},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.00405},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Multiple ergodic averages for commuting multiplicative actions
AU - Dimitrios Charamaras
AU - Andreas Koutsogiannis
AU - Konstantinos Tsinas
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.00405
UR - https://arxiv.org/abs/2609.00405
AB - We study the convergence of multiple ergodic averages involving several commuting multiplicative actions. We prove that for finitely generated systems, the averages 1/N∑_n=1^N S_1,nF_1·…· S_ℓ,nF_ℓ 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.
ER -