Periodic Joint Co-tiles of Independent Tiles
Abstract
For every d≥2, we prove that d−1 independent finite tiles in Zd admit a fully periodic joint co-tile whenever they admit a joint co-tile. Here independence means that the tiles contain the origin and every choice of one nonzero vector from each tile is linearly independent. This removes property (⋆) from the periodic existence theorem of Meyerovitch, Sanadhya and Solomon and recovers Bhattacharya's theorem when d=2. More generally, we prove periodic existence for any finite family admitting a joint co-tile whose indicator is a fully periodic real function plus finitely many bounded real functions, each invariant under a subgroup of rank d−1. Extending the Boolean normal-form method of Greenfeld and Tao, we obtain a configuration in the translation orbit closure whose restrictions to cosets of a finite-index lattice are either invariant under a rank-(d−1) subgroup or given by a three-term affine fractional-part formula with density one half. The tiling equations pair the restrictions given by this formula and determine a finite bipartite graph. A two-coloring replaces those restrictions by constants while preserving every tiling equation. We deduce a characterization of periodic tilability by independent companions and a decision algorithm for families containing an independent (d−1)-subfamily, with the dimension as part of the input.
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BibTeX
@misc{tan2026periodic,
title = {{Periodic Joint Co-tiles of Independent Tiles}},
author = {Hu Tan and Ying Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.00099},
eprint = {2610.00099},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.00099},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Periodic Joint Co-tiles of Independent Tiles
AU - Hu Tan
AU - Ying Zhang
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.00099
UR - https://arxiv.org/abs/2610.00099
AB - For every d≥2, we prove that d−1 independent finite tiles in Z^d admit a fully periodic joint co-tile whenever they admit a joint co-tile. Here independence means that the tiles contain the origin and every choice of one nonzero vector from each tile is linearly independent. This removes property (⋆) from the periodic existence theorem of Meyerovitch, Sanadhya and Solomon and recovers Bhattacharya's theorem when d=2. More generally, we prove periodic existence for any finite family admitting a joint co-tile whose indicator is a fully periodic real function plus finitely many bounded real functions, each invariant under a subgroup of rank d−1. Extending the Boolean normal-form method of Greenfeld and Tao, we obtain a configuration in the translation orbit closure whose restrictions to cosets of a finite-index lattice are either invariant under a rank-(d−1) subgroup or given by a three-term affine fractional-part formula with density one half. The tiling equations pair the restrictions given by this formula and determine a finite bipartite graph. A two-coloring replaces those restrictions by constants while preserving every tiling equation. We deduce a characterization of periodic tilability by independent companions and a decision algorithm for families containing an independent (d−1)-subfamily, with the dimension as part of the input.
ER -