Leptin densities in amenable groups
amenability, Følner net, Beurling density, Banach density, model set, almost periodicity
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.
Pogorzelski, F., Richard, C., & Strungaru, N. (2022). Leptin Densities in Amenable Groups. Journal of Fourier Analysis and Applications, 28, 85. https://doi.org/10.1007/s00041-022-09978-8
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