{
  "abstract": "We study sets of (measurable) returns in countable groups $G$, namely sets of the form $\\{g\\in G:μ(A\\cap T_gA)>0\\}$ arising from measure-preserving actions. Extending a result of Bergelson, we show that sets of returns in $G\\times G$ contain subsets of the form $B\\times B$, 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 $A\\subseteq G\\times G$ satisfies $B\\times B\\subseteq AA^{-1}$ for some large set $B\\subseteq G$. 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 $g\\mapsto μ(T_g^{-1}A\\cap T_gA)$. 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 $A\\subseteq G$ contains a large subset $B$ satisfying $BB\\subseteq AA^{-1}$. Finally, we establish polynomial analogues of these results for finitely generated nilpotent groups, extending earlier work of Bergelson and Ruzsa.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.02873",
  "author_total": 2,
  "authors": [
    {
      "name": "Vitaly Bergelson"
    },
    {
      "name": "Saúl Rodríguez-Martín"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bqhtp4000k6gh.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.02873",
  "fulltext_url": null,
  "id": "bqhtp4000k6gh",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.02873",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.DS"
    ],
    [
      "arxiv",
      "math.GR"
    ],
    [
      "openalex",
      "2610|Mathematical Dynamics and Fractals"
    ]
  ],
  "openalex": "W7172518306",
  "pdf_url": "https://arxiv.org/pdf/2608.02873",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bqhtp4000k6gh.ris",
  "sources": [
    {
      "first_seen": "2026-11-05T15:21:00Z",
      "native_id": "2608.02873",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.02873"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    }
  ],
  "title": "Product sets in sets of returns and positivity of symmetric ergodic averages",
  "title_text": "Product sets in sets of returns and positivity of symmetric ergodic averages",
  "type": "preprint",
  "updated_at": "2026-11-09T00:20:25Z",
  "url": "https://research.draconitic.com/paper/bqhtp4000k6gh",
  "venue": "arXiv",
  "versions": []
}