{
  "abstract": "Let $\\mathcal P$ be a finite nonempty set of primes, and let $S_{\\mathcal P}$ be the multiplicative semigroup that they generate. We prove that there are an invertible weakly mixing measure-preserving transformation $T$ and a set $A$ of positive measure such that $A\\cap T^{-r}A=\\varnothing$ for every $r\\in S_{\\mathcal P}$. This gives a negative answer to Bergelson's 1996 question concerning recurrence along $\\{2^m3^n:m,n\\geq1\\}$, and applies to every finitely generated multiplicative semigroup of positive integers. More generally, we give a sufficient condition for weakly mixing non-recurrence based on divisibility properties and separation by finitely many rotations of the circle. This criterion also applies to an infinitely generated multiplicative semigroup not contained in any $S_{\\mathcal P}$ with $\\mathcal P$ finite.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.33710",
  "author_total": 1,
  "authors": [
    {
      "name": "Zhuowei Liu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/b63bwak31pmpz.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.33710",
  "fulltext_url": null,
  "id": "b63bwak31pmpz",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.33710",
  "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.DS"
    ],
    [
      "openalex",
      "2610|Mathematical Dynamics and Fractals"
    ]
  ],
  "openalex": "W7224025158",
  "pdf_url": "https://arxiv.org/pdf/2610.33710",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/b63bwak31pmpz.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.33710",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.33710"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    }
  ],
  "title": "Weak mixing and non-recurrence along finitely generated multiplicative semigroups of $\\mathbb N$",
  "title_text": "Weak mixing and non-recurrence along finitely generated multiplicative semigroups of ℕ",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/b63bwak31pmpz",
  "venue": "arXiv",
  "versions": []
}