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arXiv · 2609.34852

Weak Tiling by Unions of Non-overlapping Unit Cubes

Tianyu Chen·Shilei Fan·Mihail N. Kolountzakis·Chun-Kit Lai
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Abstract

We study weak tiling by finite unions of pairwise non-overlapping unit cubes in $\mathbb R^d$ that are not necessarily axis-parallel. We obtain a weak-tiling analogue of Keller's classical theorem, which gives a structural restriction on the support of any weak tiling measure associated with the unit cube. This allows us to derive geometric restrictions on configurations of cubes whose union admits a weak tiling. As an application, we prove Fuglede's conjecture for unions of three non-overlapping unit squares in $\mathbb R^2$ as well as unions of two non-overlapping axis-parallel unit cubes in any dimension.

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Tianyu Chen, Shilei Fan, Mihail N. Kolountzakis, Chun-Kit Lai. 2026-09-28. Weak Tiling by Unions of Non-overlapping Unit Cubes. https://arxiv.org/abs/2609.34852

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