{
  "abstract": "We introduce new techniques for determining combinatorial properties of sets of multiple recurrence by considering weighted averages with quickly growing weights. Our main result is a far-reaching generalization of Szemerédi's Theorem which additionally confirms a conjecture of Bergelson-Moreira-Richter and contains as special cases both the Polynomial Szemerédi Theorem due to Bergelson-Leibman-Lesigne and the fact that if $f$ belongs to a broad class of smooth functions and satisfies $x^{d-1}\\prec f(x)\\prec x^d$ for some $d\\in \\mathbb{N}$ then for any $\\ell\\in \\mathbb{N}$, any invertible measure preserving system $(X,\\mathscr{B},μ,T)$, and any $A\\in \\mathscr{B}$ with $μ(A)>0$, the set $\\{n\\in \\mathbb{N}: μ(A\\cap T^{-[f(n)]}A\\cap T^{-2[f(n)]}A\\cap \\cdots\\cap T^{-\\ell[f(n)]}A )>0\\}$ is thick, meaning that it contains arbitrarily long intervals of natural numbers. Additionally, we formulate and prove a generalization to weighted averages of Boshernitzan's criterion for uniform distribution which we use in the proof of our main result.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.16554",
  "author_total": 2,
  "authors": [
    {
      "name": "Vitaly Bergelson"
    },
    {
      "name": "Michael Reilly"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/472r2d6adv26w.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.16554",
  "fulltext_url": null,
  "id": "472r2d6adv26w",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.16554",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.DS"
    ],
    [
      "openalex",
      "2610|Mathematical Dynamics and Fractals"
    ]
  ],
  "openalex": "W7213437528",
  "pdf_url": "https://arxiv.org/pdf/2609.16554",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/472r2d6adv26w.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:45Z",
      "native_id": "2609.16554",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.16554"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    }
  ],
  "title": "Weighted averages and applications to sets of multiple recurrence",
  "title_text": "Weighted averages and applications to sets of multiple recurrence",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:45Z",
  "url": "https://research.draconitic.com/paper/472r2d6adv26w",
  "venue": "arXiv",
  "versions": []
}