Characterising linear repetitivity for polytopal cut and project sets

Abstract

A cut and project set is given by cutting an irrational slice of a higher dimensional lattice and then projecting it to the so-called physical space. They make for interesting models of quasicrystals, since they can be designed to have symmetries forbidden for periodic patterns yet, for reasonable windows, have pure point diffraction. Two of the most fundamental measures of order for aperiodic patterns are complexity (the growth rate in the number of patches of a given size) and repetitivity (how regularly patches repeat within the pattern). For polytopal cut and project sets – those with Euclidean physical and internal space and polytopal window – these properties may be analysed via an interesting blend of discrete geometry and Diophantine approximation. I will present recent work, joint with Henna Koivusalo, that characterises linear repetitivity (LR) for a natural class of polytopal cut and project sets, extending previous results joint with Henna Koivusalo and Alan Haynes. We show that LR is equivalent to properties C and D for this class. Here, C is a low complexity condition, which itself has a simple characterisation in terms of ranks of intersections of the projected lattice with the supporting subspaces of the window. The property D is a Diophantine condition, analogous to a number being badly approximable by rationals.