Linearly repetitive Delone sets Delone sets with finite local complexity: Linear repetitivity versus positivity of weights

TL;DR

This paper explores the relationship between linear repetitivity and positivity of weights in Delone sets with finite local complexity, providing a characterization of when a subadditive ergodic theorem holds.

Contribution

It establishes that linear repetitivity in Delone sets is equivalent to the positivity of weights and balancedness of return patterns, linking geometric and ergodic properties.

Findings

Linear repetitivity is equivalent to positivity of weights.

Positivity of weights combined with balanced return patterns characterizes linear repetitivity.

A subadditive ergodic theorem holds under conditions of positive weights.

Abstract

We consider Delone sets with finite local complexity. We characterize validity of a subadditive ergodic theorem by uniform positivity of certain weights. The latter can be considered to be an averaged version of linear repetitivity. In this context, we show that linear repetitivity is equivalent to positivity of weights combined with a certain balancedness of the shape of return patterns.