Combinatorial and harmonic-analytic methods for integer tilings

TL;DR

This paper introduces new harmonic-analytic and combinatorial tools to study integer tilings, providing criteria for tiling and cyclotomic divisibility, and applying these to classify tilings with specific periods.

Contribution

It develops a systematic approach using the box product, multiscale cuboids, and saturating spaces to analyze and simplify integer tilings, advancing the theoretical understanding.

Findings

New criteria for tiling and cyclotomic divisibility

Ability to determine tiling possibilities with partial information

Reduction techniques for simplifying tiling structures

Abstract

A finite set of integers $A$ tiles the integers by translations if $\mathbb{Z}$ can be covered by pairwise disjoint translated copies of $A$. 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 with $A$ and $B$. 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 spaces, 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 set $A$ containing certain configuration 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 a follow-up paper that all tilings of period $(pqr)^2$, where $p,q,r$ are distinct odd primes, satisfy a tiling condition proposed by Coven and Meyerowitz.