{
  "abstract": "Two basic problems in the theory of integral self-affine tiles are to characterize the digit sets that generate translational tiles and to understand whether such tiles admit periodic translational tilings. We give a complete solution to the first problem: for an expanding integer matrix $\\mathbf{A}$ and a digit set $\\mathcal{D}\\subset\\mathbb{Z}^n$ with $\\#\\mathcal{D}=|\\det\\mathbf{A}|$, the self-affine set $T(\\mathbf{A},\\mathcal{D})$ is a tile if and only if, for some $N\\geq1$, the iterated digit set $\\mathcal{D}_{\\mathbf{A},N}$ is of skew-product form with respect to $\\mathbf{A}^N$. As a consequence, every integral self-affine tile admits a fully periodic self-replicating $q$-fold tiling. Moreover, $q$ divides $|T(\\mathbf{A},\\mathcal{D})|$. In dimension two we prove more: every integral self-affine tile admits a fully periodic tiling. In particular, the periodic tiling conjecture holds for integral self-affine tiles in $\\mathbb{R}^2$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.16340",
  "author_total": 2,
  "authors": [
    {
      "name": "Li-Xiang An"
    },
    {
      "name": "Ting-Ting He"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7ht1jkptyv1xt.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.16340",
  "fulltext_url": null,
  "id": "7ht1jkptyv1xt",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.16340",
  "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"
    ],
    [
      "arxiv",
      "math.FA"
    ],
    [
      "openalex",
      "2610|Mathematical Dynamics and Fractals"
    ]
  ],
  "openalex": "W7221806002",
  "pdf_url": "https://arxiv.org/pdf/2610.16340",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7ht1jkptyv1xt.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2610.16340",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.16340"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Translational self-affine tiles in $\\mathbb{R}^n$",
  "title_text": "Translational self-affine tiles in ℝ^n",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/7ht1jkptyv1xt",
  "venue": "arXiv",
  "versions": []
}