  Starting out from results known for the most classical cases of N, Z^d, R^d
or for sigma-finite abelian groups, here we define the notion of asymptotic
uniform upper density in general locally compact abelian groups. Even if a bit
surprising, the new notion proves to be the right extension of the classical
cases of Z^d, R^d. The new notion is used to extend some analogous results
previously obtained only for classical cases or sigma-finite abelian groups. In
particular, we show the following extension of a well-known result for Z of
Furstenberg: if in a general locally compact Abelian group G a subset S of G
has positive uniform asymptotic upper density, then S-S is syndetic.
