Linear repetitivity and subadditive ergodic theorems for cut and project sets

TL;DR

This paper characterizes linearly repetitive cut and project sets with cubical windows using algebraic, geometric, and dynamical methods, linking them to subadditive ergodic theorems and enabling applications to classical models.

Contribution

It provides a complete characterization of linearly repetitive cut and project sets with cubical windows and establishes their equivalence with sets satisfying subadditive ergodic theorems.

Findings

Characterization of all linearly repetitive cut and project sets with cubical windows.

Proof that these sets satisfy subadditive ergodic theorems.

Construction of such sets in various dimensions and codimensions.

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.