Translational self-affine tiles in ℝⁿ

@misc{an2026translational,
  title = {{Translational self-affine tiles in $\mathbbR^n$}},
  author = {Li-Xiang An and Ting-Ting He},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.16340},
  eprint = {2610.16340},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.16340},
}

TY  - UNPB
TI  - Translational self-affine tiles in ℝ^n
AU  - Li-Xiang An
AU  - Ting-Ting He
PY  - 2026
DA  - 2026/10/12
PB  - arXiv
DO  - 10.48550/arxiv.2610.16340
UR  - https://arxiv.org/abs/2610.16340
AB  - 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 A and a digit set 𝒟⊂ℤ^n with #𝒟=|detA|, the self-affine set T(A,𝒟) is a tile if and only if, for some N≥1, the iterated digit set 𝒟_A,N is of skew-product form with respect to A^N. As a consequence, every integral self-affine tile admits a fully periodic self-replicating q-fold tiling. Moreover, q divides |T(A,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 ℝ^2.
ER  -