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arXiv · 2610.00099

Periodic Joint Co-tiles of Independent Tiles

Hu Tan·Ying Zhang
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Abstract

For every $d\geq2$, 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 $(\star)$ 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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Hu Tan, Ying Zhang. 2026-09-08. Periodic Joint Co-tiles of Independent Tiles. https://arxiv.org/abs/2610.00099

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