{
  "abstract": "We prove that, for commuting unitary representations $U,V$ of a countable amenable group $G$, the averages $$ D_F=\\frac1{|F|}\\sum_{g\\in F}U_g^*V_g $$ have a common selfadjoint weak operator limit along all left and right \\Folner{} sequences, with strong* convergence along two-sided sequences. This answers Bergelson's question on commuting actions. For any unitary representation $π$, we also prove that $A_F=|F|^{-1}\\sum_{g\\in F}π(g^2)$ converges strongly along right \\Folner{} sequences to a selfadjoint central operator supported on the finite type~I part of $π(G)''$; the adjoints $A_F^*$ converge strongly along left \\Folner{} sequences. This settles the convergence question for square averages posed by Bergelson and Rodríguez-Martín. For measure-preserving actions of finitely generated amenable or countable virtually FC groups, we obtain $$ \\lim_n\\frac1{|F_n|}\\sum_{g\\in F_n}μ(A\\cap T_{g^2}A)>0 \\qquad\\text{whenever }μ(A)>0. $$ For each $\\varepsilon>0$, the set $\\{g:μ(A\\cap T_{g^2}A)>μ(A)^2-\\varepsilon\\}$ meets every Bohr neighborhood of the identity in a set of positive lower density, uniformly over \\Folner{} sequences. These intersections are also left and right IP$^*$ for virtually FC groups. We construct a unitary representation $π$ of a locally finite nilpotent group and a weakly mixing vector $f$ for which $\\{g:\\text{Re}\\langleπ(g^2)f,f\\rangle>-\\varepsilon\\}$ has upper Banach density zero for small $\\varepsilon>0$. We also give a weakly mixing action of a two-generated solvable group for which the IP$^*$ conclusion fails. The recurrence results yield product sets $BB$ inside difference sets, with $B$ of positive upper density.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.02901",
  "author_total": 1,
  "authors": [
    {
      "name": "Xiaofang Luo"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dc7gevpzzq6v6.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.02901",
  "fulltext_url": null,
  "id": "dc7gevpzzq6v6",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.02901",
  "language": null,
  "licence": "CC-BY-NC-SA-4.0",
  "licence_label": "CC BY-NC-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.DS"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.02901",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dc7gevpzzq6v6.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:14Z",
      "native_id": "2611.02901",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.02901"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    }
  ],
  "title": "Antidiagonal and square averages for amenable groups",
  "title_text": "Antidiagonal and square averages for amenable groups",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/dc7gevpzzq6v6",
  "venue": "arXiv",
  "versions": []
}