Combinatorial and harmonic-analytic methods for integer tilings

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.

Keywords

• 05B45
• 11B75
• 20K0
• 11C08
• 43A75
• 51D20
• 52C22