Indecomposable Tilings of the Integers with Exponentially Long Periods

Abstract

Let $A$ be a finite multiset of integers. A second multiset of integers $T$ is said to be an $A$-tiling of level $d$ if every integer can be expressed in exactly $d$ ways as the sum of an element of $A$ and of an element of $T$. The set $T$ is indecomposable if it cannot be written as the disjoint union of two proper subsets that are also $A$-tilings. In this paper we show how to construct indecomposable tilings that have exponentially long periods. More precisely, we give a sequence of multisets $(A_k)_{k=1}^{\infty}$ such that each $A_k$ admits an indecomposable tiling $T_k$ of period greater than $e^{c\root 3\of{n_k\log(n_k)}}$ where $n_k = {\rm diam}(A_k) = \max\{j \in A_k\} - \min\{j \in A_k\}$ tends to infinity and where $c > 0$ is some constant independent of $k$.