Tilings of $\mathbb{Z}$ with multisets of distances

Abstract

In this paper, we study tilings of $\mathbb{Z}$, that is, coverings of $\mathbb{Z}$ by disjoint sets (tiles). Let $T=\{d_1,\dots ,d_s\}$ be a given multiset of distances. Is it always possible to tile $\mathbb{Z}$ by tiles, for which the multiset of distances between consecutive points is equal to $T$? In this paper, we give a sufficient condition that such a tiling exists. Our result allows multisets of distances to have arbitrarily many distinct values. Our result generalizes most of the previously known results, all of which dealt with the cases of $2$ or $3$ distinct distances.