Tools for investigating recurrence properties of polytopal cut and project schemes

Abstract

I will give an overview of the various techniques used to understand how many patches of a certain size may be found in a polytopal cut and project set (its “complexity”) and how frequently those patches recur (its “recurrence”, or “repetitivity”). In recent work with Henna Koivusalo we defined a new property for polytopal cut and project schemes, the approximately canonical condition, which allows one to replace the analysis of acceptance domains of patches with a simpler analysis via so-called cut regions. I will explain through an example why this property is essentially the minimal one necessary for this purpose by showing that the slightly weaker almost canonical condition, which is in standard usage, does not allow one to directly replace acceptance domains with cut regions. I will go into further detail on why a Diophantine condition for the scheme, in combination with minimal complexity, is necessary and sufficient for the associated cut and project sets to be linearly repetitive.