Combinatorial and harmonic-analytic methods for integer tilings

Abstract

AbstractA finite set of integersAtiles the integers by translations if$\mathbb {Z}$can be covered by pairwise disjoint translated copies ofA. Restricting attention to one tiling period, we have$A\oplus B=\mathbb {Z}_M$for some$M\in \mathbb {N}$and$B\subset \mathbb {Z}$. This can also be stated in terms of cyclotomic divisibility of the mask polynomials$A(X)$and$B(X)$associated withAandB.In this article, we introduce a new approach to a systematic study of such tilings. Our main new tools are the box product, multiscale cuboids and saturating sets, developed through a combination of harmonic-analytic and combinatorial methods. We provide new criteria for tiling and cyclotomic divisibility in terms of these concepts. As an application, we can determine whether a setAcontaining certain configurations can tile a cyclic group$\mathbb {Z}_M$, or recover a tiling set based on partial information about it. We also develop tiling reductions where a given tiling can be replaced by one or more tilings with a simpler structure. The tools introduced here are crucial in our proof in [24] that all tilings of period$(pqr)^2$, where$p,q,r$are distinct odd primes, satisfy a tiling condition proposed by Coven and Meyerowitz [2].