{
  "abstract": "Let $(X,\\mathcal B,μ)$ be a probability space and let $S,Q:X\\to X$ be commuting probability-preserving transformations. Let $1\\leq p,q\\leq\\infty$ satisfy $1/p+1/q\\leq1$. We prove that, for every fixed $F\\in L^p(μ)$ and $G\\in L^q(μ)$, the averages $$ \\frac1N\\sum_{n=1}^N F(S^nx)G(Q^nx) $$ converge almost everywhere as $N\\to\\infty$. Our proof introduces a new analytic approach that relies on neither the circle method nor characteristic-factor theory. Exploiting the triangular structure arising from commuting transformations, we combine matrix-energy estimates with Gaussian heat-flow arguments to establish pointwise variation inequalities for suitable smooth averages. We then use even and odd Gaussian kernels to recover the convergence of the original averages.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.26388",
  "author_total": 3,
  "authors": [
    {
      "name": "Zhuowen Guo"
    },
    {
      "name": "Rongzhong Xiao"
    },
    {
      "name": "Shuhao Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6hg3skrwzs04r.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.26388",
  "fulltext_url": null,
  "id": "6hg3skrwzs04r",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.26388",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.DS"
    ],
    [
      "openalex",
      "2610|Mathematical Dynamics and Fractals"
    ]
  ],
  "openalex": "W7223003786",
  "pdf_url": "https://arxiv.org/pdf/2610.26388",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6hg3skrwzs04r.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2610.26388",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.26388"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    }
  ],
  "title": "Pointwise convergence of bilinear ergodic averages for commuting transformations",
  "title_text": "Pointwise convergence of bilinear ergodic averages for commuting transformations",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/6hg3skrwzs04r",
  "venue": "arXiv",
  "versions": []
}