Cut and project sets are regular, but nevertheless aperiodic point patterns obtained by projecting an irrational slice of the integer lattice to a subspace. There is a flexible formalism for translating information on Diophantine approximation to regularity properties of cut and project sets. In this talk I will start from the definition of cut and project sets and the history of studying them. Then I will explain how to use repetitivity as a way of measuring regularity of cut and project sets and, in particular, how to quantify repetitivity properties of cut and project sets using Diophantine approximation.
The talk is based on several joint works, with subsets of Alan Haynes, Antoine Julien and Jamie Walton.