Unimodular Billingsley and Frostman Lemmas

TL;DR

This paper develops unimodular versions of classical geometric measure theory lemmas, linking unimodular dimensions to growth rates and providing tools to analyze random discrete metric spaces.

Contribution

It introduces unimodular analogues of Billingsley's lemma, Frostman's lemma, and the mass distribution principle, connecting unimodular dimensions with growth and equivariant measures.

Findings

Unimodular versions of classical measure lemmas are established.

Upper bounds on unimodular Hausdorff dimension are derived from growth rates.

The results apply to point processes, unimodular graphs, and self-similar structures.

Abstract

The notions of unimodular Minkowski and Hausdorff dimensions are defined in [arXiv:1807.02980] for unimodular random discrete metric spaces. The present paper is focused on the connections between these notions and the polynomial growth rate of the underlying space. It is shown that bounding the dimension is closely related to finding suitable equivariant weight functions (i.e., measures) on the underlying discrete space. The main results are unimodular versions of the mass distribution principle, Billingsley's lemma and Frostman's lemma, which allow one to derive upper bounds on the unimodular Hausdorff dimension from the growth rate of suitable equivariant weight functions. These results allow one to compute or bound both types of unimodular dimensions in a large set of examples in the theory of point processes, unimodular random graphs, and self-similarity. Further results of independent interest are also presented, like a version of the max-flow min-cut theorem for unimodular one-ended trees and a weak form of pointwise ergodic theorems for all unimodular discrete spaces.