Multiple tilings of Z with long periods, and tiles with many-generated level semigroups
John P. Steinberger · 2005
We consider multiple tilings of Z by translates of a finite multiset A of integers (called a tile). We say that a set of integers T is an A-tiling of level d if each integer can be written in exactly d ways as the sum of an element of T and an element of A. We find new exponential lower bounds on the longest period of A-tiling as a function of the diameter of A, which rejoin the exponential upper bounds given by Ruzsa (preprint, 2002) and Kolountza- kis (2003). We also show the existence of tiles whose level semigroups have arbitrarily many generators (where the level semigroup of a tile A is the set of integers d such that A admits a tiling of level d).