Linear repetitivity for polytopal cut and project sets

Abstract

One of the strongest regularity properties an aperiodic pattern can have is of being linearly repetitive, which means that finite sub-patterns are spaced in a highly regular manner across the pattern leaving only small gaps between them. Substitution tilings (under standard restrictions) always have linear repetitivity but the question for cut and project sets is more difficult. In this talk I will give a simple introduction to both the cut and project method and the property of linear repetitivity. After this I will explain recent joint results with Henna Koivusalo on characterising linear repetitivity for cut and project sets whose windows are convex polytopes, which connects the dynamics of aperiodic patterns with the branch of Number Theory called Diophantine Approximation.