{
  "abstract": "Varjú and Yu have given a complete description of non-singleton self-similar sets of uniqueness on the real line $\\mathbb R$. In this paper, we first prove an arithmetic product theorem for self-similar $U$-sets $E:$ if $F\\subset\\mathbb R$ is compact, then $F\\cdot E$ is an $M_0$-set if and only if $F$ is uncountable. We next show that this conclusion remains valid for bounded integer-based homogeneous Moran sets. As an application, for every $s\\in(0,1]$, there exists a bounded integer-based homogeneous Moran set of uniqueness $K$ such that $\\dim_H K=s$ and the self-product $K\\cdot K$ is an $M_0$-set. In sharp contrast, for every $s\\in[0,1/2]$ we can construct an unbounded integer-based homogeneous Moran $U$-set $E$ with $\\dim_H E=s$ such that both $E$ and $E\\cdot E$ are $U$-sets, and $1/2$ is the optimal threshold for $U$-sets to be closed under the arithmetic self-product. Moreover, we show that if $E\\subset[0,1]$ is a compact set whose self-product $E\\cdot E$ is contained in an $H$-set, then $\\underline\\dim_B E\\le1/2$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.04956",
  "author_total": 3,
  "authors": [
    {
      "name": "Yan-Song Fu"
    },
    {
      "name": "Xu-Guang Wang"
    },
    {
      "name": "Meng Zhu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/aq8549zb5yepr.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.04956",
  "fulltext_url": null,
  "id": "aq8549zb5yepr",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.04956",
  "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.CA"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.04956",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/aq8549zb5yepr.ris",
  "sources": [
    {
      "first_seen": "2026-11-04T15:21:29Z",
      "native_id": "2611.04956",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.04956"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Arithmetic product theorems on self-similar sets of uniqueness and homogeneous Moran sets",
  "title_text": "Arithmetic product theorems on self-similar sets of uniqueness and homogeneous Moran sets",
  "type": "preprint",
  "updated_at": "2026-11-06T18:21:01Z",
  "url": "https://research.draconitic.com/paper/aq8549zb5yepr",
  "venue": "arXiv",
  "versions": []
}