Linear repetitivity and subadditive ergodic theorems for cut and project sets

2015/03/13 by Haynes, Alan, Koivusalo, Henna, Walton, James
#Dynamical Systems (math.DS) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1503.04091

Abstract

By using a combination of algebraic, geometric, and dynamical techniques, together with input from higher dimensional Diophantine approximation, we give a complete characterization of all linearly repetitive cut and project sets with cubical windows. We also prove that these are precisely the collection of such sets which satisfy subadditive ergodic theorems. The results are explicit enough to allow us to apply them to known classical models, and to construct linearly repetitive cut and project sets in all pairs of dimensions and codimensions in which they exist.

Related

• Cut and project sets with polytopal window II: linear repetitivity
• Perfectly ordered quasicrystals and the Littlewood conjecture
• A characterisation of linear repetitivity for cut and project sets with general polytopal windows
• Linear repetitivity beyond abelian groups
• Model sets, almost periodic patterns, uniform density and linear maps