Patch frequencies for cut-and-project sets & Diophantine approximation

Abstract

The cut-and-project scheme is a method of producing interesting point sets of Euclidean space, such as the vertex sets of the famous Penrose tilings. Cut-and-project sets are important examples in the field of aperiodic order, the study of long-range geometric order in objects which do not necessarily possess global translational symmetry. For example, these point sets have pure point diffraction, which makes them good models for quasicrystals. In this talk, I shall discuss recent collaborative work with Alan Haynes, Henna Koivusalo and Lorenzo Sadun, in which we show that for substantial collections of cut-and-project sets the numbers of distinct frequencies of subpatches of size r remains very low, or even bounded, as r tends to infinity. These results are proved by connecting frequency spectrums of cut-and-project sets to ‘gaps problems’ in Diophantine approximation.