可顺从群中的Leptin密度
Leptin densities in amenable groups
AI总结：
本文基于Leptin群不变量在局部紧群上定义均匀密度，证明其在幺模可顺从群中与Beurling密度一致，并用其几何证明模型集密度公式及均匀平均几乎周期性。
摘要：
考虑局部紧群上的正Borel测度。我们为这类测度定义了一种均匀密度的概念，该概念基于Leptin于1966年引入的群不变量。随后我们限制在幺模可顺从群和平移有界测度的情形。在这种情况下，我们的密度概念与傅里叶分析中著名的Beurling密度（在动力系统理论中也称为Banach密度）一致。我们利用Leptin密度给出模型集密度公式的几何证明，该公式用均匀正则模型集的窗口体积来表示其密度，并用于证明此类模型集的均匀平均几乎周期性。
英文摘要：
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.
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