Polytopal cut and project sets: complexity and repetitivity

Abstract

I will explain recent work with Henna Koivusalo which investigates the combinatorial complexity and recurrence of Delone sets constructed by the cut and project method with polytopal windows. The complexity concerns the asymptotic growth rate of the number of radius r patches as r tends to infinity. We introduce the ‘quasicanonical’ condition, which allows one to replace acceptance domains with regions cut by translated hyperplanes, a property which we show fails to hold in general under the ‘almost canonical’ condition. Using a more refined argument, we also derive a formula for the asymptotic growth rate of the complexity function without assuming this condition, generalising a result of Julien. We characterise linear repetitivity for a broad class of polytopal cut and project sets, including those with indecomposable and canonical window. I will summarise some remaining subtle obstacles to a more general characterisation which removes the indecomposable assumption.