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.