Linear repetitivity for polytopal cut and project sets

Abstract

I will discuss recent work with Henna Koivusalo that concerns the question of which Euclidean cut and project schemes with polytopal windows lead to patterns that are linearly repetitive. This builds upon previous work with Haynes and Koivusalo resolving the question for so-called ‘cubical windows’. We find a necessary and sufficient condition for linear repetitivity which applies to a large class of schemes satisfying a minor natural strengthening of the standard ‘almost canonical’ property. The condition, and the proof of equivalence with linear repetitivity, involves an interesting blend of discrete geometry and Diophantine approximation.