 

John P. Steinberger
Multiple tilings of Z with long periods, and tiles with manygenerated level semigroups


Published: 
September 22, 2005 
Keywords: 
tilings, level semigroups, weight semigroups, multiple tilings, period, periods, periodicity 
Subject: 
11B13, 05B45 


Abstract
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 Atiling 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 Atiling
as a function of the diameter of A, which rejoin the exponential upper bounds given by
Ruzsa (preprint, 2002) and Kolountzakis (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).


Author information
Department of Mathematics, UC Davis
jpsteinb@math.ucdavis.edu
http://www.math.ucdavis.edu/~jpsteinb/index.html

