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arXiv · 2010.03254

The structure of translational tilings in $\mathbb{Z}^d$

Rachel Greenfeld·Terence Tao
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Abstract

We obtain structural results on translational tilings of periodic functions in $\mathbb{Z}^d$ by finite tiles. In particular, we show that any level one tiling of a periodic set in $\mathbb{Z}^2$ must be weakly periodic (the disjoint union of sets that are individually periodic in one direction), but present a counterexample of a higher level tiling of $\mathbb{Z}^2$ that fails to be weakly periodic. We also establish a quantitative version of the two-dimensional periodic tiling conjecture which asserts that any finite tile in $\mathbb{Z}^2$ that admits a tiling, must admit a periodic tiling, by providing a polynomial bound on the period; this also gives an exponential-type bound on the computational complexity of the problem of deciding whether a given finite subset of $\mathbb{Z}^2$ tiles or not. As a byproduct of our structural theory, we also obtain an explicit formula for a universal period for all tilings of a one-dimensional tile.

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Rachel Greenfeld, Terence Tao. 2020-10-07. The structure of translational tilings in $\mathbb{Z}^d$. https://arxiv.org/abs/2010.03254

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