Translational tilings by axis-parallel polygonal sets

Abstract

The periodic tiling conjecture asserts that if a bounded set Ω⊂ ℝ^d of finite measure tiles ℝ^d by translations, then it admits at least one fully periodic tiling. This conjecture was proved to hold in ℝ by Lagarias and Wang, and was recently disproved by Greenfeld and Tao in sufficiently high dimensions. Its discrete analogue in ℤ² was proved by Bhattacharya, whereas the conjecture in ℝ² remains open. In this paper, we analyze the structure of tilings by axis-parallel polygonal sets: bounded open sets in ℝ² 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 ℝ² 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.

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BibTeX RIS

BibTeX

@misc{pont2026translational,
  title = {{Translational tilings by axis-parallel polygonal sets}},
  author = {Jaume de Dios Pont and Jan Grebík and Rachel Greenfeld and José Madrid},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.15785},
  eprint = {2610.15785},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.15785},
}

RIS

TY  - UNPB
TI  - Translational tilings by axis-parallel polygonal sets
AU  - Jaume de Dios Pont
AU  - Jan Grebík
AU  - Rachel Greenfeld
AU  - José Madrid
PY  - 2026
DA  - 2026/10/11
PB  - arXiv
DO  - 10.48550/arxiv.2610.15785
UR  - https://arxiv.org/abs/2610.15785
AB  - The periodic tiling conjecture asserts that if a bounded set Ω⊂ ℝ^d of finite measure tiles ℝ^d by translations, then it admits at least one fully periodic tiling. This conjecture was proved to hold in ℝ by Lagarias and Wang, and was recently disproved by Greenfeld and Tao in sufficiently high dimensions. Its discrete analogue in ℤ^2 was proved by Bhattacharya, whereas the conjecture in ℝ^2 remains open. In this paper, we analyze the structure of tilings by axis-parallel polygonal sets: bounded open sets in 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 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.
ER  -

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