Linear repetitivity of cut-and-project sets

Abstract

The two main sources of aperiodically ordered patterns are the cut-and-project method and tiling substitutions. Some questions are simple for patterns coming from one of these constructions but are difficult for those coming from the other. For example, it is easy to show that patterns coming from the cut-and-project method exhibit pure point diffraction, but the question is difficult for substitution tilings -the famous Pisot Conjecture on this remains unsolved. In the other direction, it is easy to show that all (primitive) substitution tilings are linearly repetitive, that is, there exists some ${C>0}$ for which every sub-patch of the pattern of size ${r}$ occurs within radius ${Cr}$ of any point of the pattern. I will discuss a recent result, generalising a classical result of Hedlund and Morse on Sturmian sequences, which states that acanonical codimension one cut-and-project set is linearly repetitive if and only if the physical space used in its construction corresponds to a badly approximable linear form.