Linear repetitivity beyond abelian groups
Abstract
We show that linearly repetitive weighted Delone sets in groups of polynomial growth have a uniquely ergodic hull. This result applies in particular to the linearly repetitive weighted Delone sets in homogeneous Lie groups constructed in the companion paper arXiv:2109.15210 using symbolic substitution methods. More generally, using the quasi-tiling method of Ornstein-Weiss, we establish unique ergodicity of hulls of weighted Delone sets in amenable unimodular lcsc groups under a new repetitivity condition which we call tempered repetitivity. For this purpose, we establish a general sub-additive convergence theorem, which also has applications concerning the existence of Banach densities and uniform approximation of the spectral distribution function of finite hopping range operators on Cayley graphs.
Cite
@article{arxiv.2001.10725,
title = {Linear repetitivity beyond abelian groups},
author = {Siegfried Beckus and Tobias Hartnick and Felix Pogorzelski},
journal= {arXiv preprint arXiv:2001.10725},
year = {2025}
}