  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.
