Henna Koivusalo (University of Bristol)

A biased take on cut and project sets — overview of definitions and some properties

Abstract:

Cut and project sets are obtained by taking an irrational slice through a lattice and projecting it to a lower dimensional subspace. This usually results in a set which has no translational period, even though it retains a lot of the regularity of the lattice. As such, cut and project sets are one of the archetypical examples of point sets featuring aperiodic order. The definition of cut and project sets allows for many interpretations and generalisations, and they can naturally be studied in the context of dynamical systems, discrete geometry, harmonic analysis, or Diophantine approximation, for example, depending on one's own tastes and interests. This talk will touch on the definition and various interpretations of cut and project sets, and cover some of my new(ish) and old(ish) results on finite pattern repetition and the like. The talk is based on ideas developed over several joint works, with coauthors such as Alan Haynes, Jamie Walton, Antoine Julian and Lorenzo Sadun.