Leptin densities in amenable groups

TL;DR

This paper introduces a new notion of uniform density for measures on locally compact groups, connecting it with classical densities and applying it to model set density and almost periodicity.

Contribution

It defines Leptin densities for measures on groups and demonstrates their equivalence to known densities in specific cases, providing new geometric and periodicity results.

Findings

Leptin densities coincide with Beurling densities on unimodular amenable groups.

The density formula for model sets is proven using Leptin densities.

Model sets exhibit uniform mean almost periodicity.

Abstract

Consider a positive Borel measure on a locally compact group. We define a notion of uniform density for such a measure, which is based on a group invariant introduced by Leptin in 1966. We then restrict to unimodular amenable groups and to translation bounded measures. In that case our density notion coincides with the well-known Beurling density from Fourier analysis, also known as Banach density from dynamical systems theory. We use Leptin densities for a geometric proof of the model set density formula, which expresses the density of a uniform regular model set in terms of the volume of its window, and for a proof of uniform mean almost periodicity of such model sets.