Patch frequencies for cut-and-project sets

Abstract

The cut-and-project scheme is an important method of constructing aperiodically ordered point sets of Euclidean space, such as the set of vertices of a Penrose tiling. In this talk I shall describe how the frequencies of finite patches of a cut-and-project set are determined by the volumes of certain regions defined by a toral translation. This was exploited in recent collaborative work with Alan Haynes, Henna Koivusalo and Lorenzo Sadun, where we showed that certain Diophantine conditions on the setup impose a strict behaviour of the frequency spectrum. From these considerations, one may give a highly geometric proof of the Three Gap Theorem (also known as The Steinhaus Theorem) and I shall discuss the possibility of these methods extending to higher dimensional settings.