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Leptin densities in amenable groups

Felix Pogorzelski, Christoph Richard, Nicolae Strungaru

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.

Leptin densities in amenable groups

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.

Paper Structure

This paper contains 29 sections, 29 theorems, 94 equations.

Table of Contents

  1. Introduction
  2. Delone measures
  3. Definition of Delone measures
  4. Standard estimates on unimodular groups
  5. Leptin density of a measure
  6. Definition
  7. Some properties of Leptin densities
  8. Leptin densities of Delone measures in unimodular groups
  9. Leptin densities and Beurling densities
  10. Leptin densities and Beurling densities
  11. Proof of Theorem \ref{['BduD1']}
  12. Følner conditions and Følner nets
  13. Various boundaries
  14. Følner nets
  15. Densities on Følner nets in unimodular groups
  16. ...and 14 more sections

Key Result

Lemma 2.3

Let $G$ be a unimodular locally compact group with Haar measure $m$. Consider any compact $A\in\mathcal{K}$ and any unit neighborhood $B$. Take $\{a_1,\ldots, a_n\}\subseteq A$ maximal such that $Ba_i\cap Ba_j=\varnothing$ for $i\ne j$. Then where the union on the lhs is disjoint. In particular we have

Theorems & Definitions (77)

  • Definition 2.1: Delone measure
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Lemma 3.1
  • proof
  • ...and 67 more