Aperiodic Order and linear repetitivity of cut and project sets

Abstract

In this talk I will introduce the theory of Aperiodic Order by showcasing some standard examples of aperiodic tilings, giving a brief overview of the field’s connection with other areas and explaining the two main construction methods: substitution and the cut and project method. In the latter, one cuts points of a lattice which fall into a thickening, by a ‘window’, of an irrational hyperplane (the ‘physical space’), and then projecting to the physical space to obtain an ordered but non-periodic point set. Many important examples may be constructed in this way, such as the vertices of the Penrose tilings and the Ammann–Beenker tilings, and the method has interesting connections to Number Theory. I will explain a recent result for Euclidean cut and project sets with polytopal windows that classifies those which are ‘the most ordered’, in the precise sense of being linearly repetitive (LR). The result is that LR is equivalent to an algebraic condition together with a Diophantine Approximation condition, of the physical space staying far from the lattice.