{
  "abstract": "The periodic tiling conjecture asserts that if a bounded set $Ω\\subset \\mathbb R^d$ of finite measure tiles $\\mathbb R^d$ by translations, then it admits at least one fully periodic tiling. This conjecture was proved to hold in $\\mathbb R$ by Lagarias and Wang, and was recently disproved by Greenfeld and Tao in sufficiently high dimensions. Its discrete analogue in $\\mathbb Z^2$ was proved by Bhattacharya, whereas the conjecture in $\\mathbb R^2$ remains open. In this paper, we analyze the structure of tilings by axis-parallel polygonal sets: bounded open sets in $\\mathbb R^2$ whose boundary is a finite union of line segments parallel to the axes. We prove that every such tiling that is topologically minimal is weakly periodic. Moreover, we show that if an axis-parallel polygonal set $Ω$ tiles $\\mathbb{R}^2$ then it admits a quasi-periodic tiling (i.e., a finite union of translated lattices). The proofs combine geometric decompositions of tilings with rational approximation and discrete tiling theory. As a consequence, we obtain that the translational tiling problem for axis-parallel polygonal sets with real algebraic vertex coordinates is decidable.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.15785",
  "author_total": 4,
  "authors": [
    {
      "name": "Jaume de Dios Pont"
    },
    {
      "name": "Jan Grebík"
    },
    {
      "name": "Rachel Greenfeld"
    },
    {
      "name": "José Madrid"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fa13nn3bw6k68.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.15785",
  "fulltext_url": null,
  "id": "fa13nn3bw6k68",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.15785",
  "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.CO"
    ],
    [
      "arxiv",
      "math.DS"
    ],
    [
      "arxiv",
      "math.LO"
    ],
    [
      "openalex",
      "2505|Quasicrystal Structures and Properties"
    ]
  ],
  "openalex": "W7221826977",
  "pdf_url": "https://arxiv.org/pdf/2610.15785",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-11",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fa13nn3bw6k68.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2610.15785",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.15785"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.LO",
      "subfield": "logic-foundations"
    }
  ],
  "title": "Translational tilings by axis-parallel polygonal sets",
  "title_text": "Translational tilings by axis-parallel polygonal sets",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/fa13nn3bw6k68",
  "venue": "arXiv",
  "versions": []
}