Leptin Densities in Amenable Groups

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 notio...

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Bibliographic Details
Published inThe Journal of fourier analysis and applications Vol. 28; no. 6
Main Authors Pogorzelski, Felix, Richard, Christoph, Strungaru, Nicolae
Format Journal Article
LanguageEnglish
Published New York Springer US 01.12.2022
Springer
Springer Nature B.V
Subjects
Abstract Harmonic Analysis
Analysis
Approximations and Expansions
Density
Dynamic systems theory
Fourier Analysis
Leptin
Mathematical Methods in Physics
Mathematics
Mathematics and Statistics
Partial Differential Equations
Signal,Image and Speech Processing
Specific gravity
78A45
Følner net
Almost periodicity
Beurling density
52C23
43A60
Banach density
Model set
Amenability
43A07
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Summary: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.
ISSN:1069-5869
1531-5851
DOI:10.1007/s00041-022-09978-8