Product sets in sets of returns and positivity of symmetric ergodic averages
Abstract
We study sets of (measurable) returns in countable groups G, namely sets of the form {g∈ G:μ(A∩ TgA)>0} arising from measure-preserving actions. Extending a result of Bergelson, we show that sets of returns in G× G contain subsets of the form B× B, where B is large with respect to suitable notions of largeness that remain meaningful even for non-amenable groups. As a consequence, if G is amenable, then every sufficiently large subset A⊆ G× G satisfies B× B⊆ AA−1 for some large set B⊆ G. We also investigate when sets of returns in G contain product sets BB with B large. In contrast with the Cartesian-product phenomenon above, this problem is considerably subtler in non-abelian groups and is closely connected to `symmetric correlation functions', namely functions of the form g↦ μ(Tg−1A∩ TgA). We use this connection to show that, for broad classes of amenable groups - including finitely generated nilpotent groups and certain solvable non-nilpotent groups, every sufficiently large set A⊆ G contains a large subset B satisfying BB⊆ AA−1. Finally, we establish polynomial analogues of these results for finitely generated nilpotent groups, extending earlier work of Bergelson and Ruzsa.
Related
- Multiple ergodic averages for commuting multiplicative actions
- Pointwise convergence of bilinear ergodic averages for commuting transformations
- Pointwise Convergence of Random Ergodic Averages at the L1 Endpoint
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BibTeX
@misc{bergelson2026product,
title = {{Product sets in sets of returns and positivity of symmetric ergodic averages}},
author = {Vitaly Bergelson and Saúl Rodríguez-Martín},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2608.02873},
eprint = {2608.02873},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.02873},
}
RIS
TY - UNPB
TI - Product sets in sets of returns and positivity of symmetric ergodic averages
AU - Vitaly Bergelson
AU - Saúl Rodríguez-Martín
PY - 2026
DA - 2026/11/04
PB - arXiv
DO - 10.48550/arxiv.2608.02873
UR - https://arxiv.org/abs/2608.02873
AB - We study sets of (measurable) returns in countable groups G, namely sets of the form {g∈ G:μ(A∩ T_gA)>0} arising from measure-preserving actions. Extending a result of Bergelson, we show that sets of returns in G× G contain subsets of the form B× B, where B is large with respect to suitable notions of largeness that remain meaningful even for non-amenable groups. As a consequence, if G is amenable, then every sufficiently large subset A⊆ G× G satisfies B× B⊆ AA^−1 for some large set B⊆ G. We also investigate when sets of returns in G contain product sets BB with B large. In contrast with the Cartesian-product phenomenon above, this problem is considerably subtler in non-abelian groups and is closely connected to `symmetric correlation functions', namely functions of the form g↦ μ(T_g^−1A∩ T_gA). We use this connection to show that, for broad classes of amenable groups - including finitely generated nilpotent groups and certain solvable non-nilpotent groups, every sufficiently large set A⊆ G contains a large subset B satisfying BB⊆ AA^−1. Finally, we establish polynomial analogues of these results for finitely generated nilpotent groups, extending earlier work of Bergelson and Ruzsa.
ER -