Periodicities of tilings of the integersAlgebra & Discrete Mathematics
|John Paul Steinberger, UC Davis
|Fri, Oct 14 2005, 1:10PM
A "translational tiling of the integers" is a an arrangement of translates of some finite set A of integers such that every integer is covered by exactly one translate of A. It has long been known that all such tilings are periodic, but the maximum length of a period, as a function of the diameter of A, remains poorly understood. We will discuss our recent result that the period of a tiling by A can grow faster than any power of the diameter of A, in other words that tilings can have superpolynomial periods. The previous best bound, due to Kolountzakis (2001), showed that the period could grow as fast as the square of the diameter of A.