Indecomposable Tilings of the Integers with Exponentially Long Periods

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 (Ak)k=1 such that each Ak admits an indecomposable tiling Tk of period greater than e 3 √ nk log(nk) where nk = diam(Ak) = max{j ∈ Ak} − min{j ∈ Ak} tends to infinity and where c > 0 is some constant independent of k.