  Asymptotic uniform upper density, shortened as a.u.u.d., or simply upper
density, is a classical notion which was first introduced by Kahane for
sequences in the real line. Syndetic sets were defined by Gottschalk and
Hendlund. For a locally compact group $G$, a set $S\subset G$ is syndetic, if
there exists a compact subset $C\Subset G$ such that $SC=G$. Syndetic sets play
an important role in various fields of applications of topological groups and
semigroups, ergodic theory and number theory. A lemma in the book of
F\"urstenberg says that once a subset $A \subset {\mathbb Z}$ has positive
a.u.u.d., then its difference set $A-A$ is syndetic. The construction of a
reasonable notion of a.u.u.d. in general locally compact Abelian groups (LCA
groups for short) was not known for long, but in the late 2000's several
constructions were worked out to generalize it from the base cases of ${\mathbb
Z}^d$ and ${\mathbb R}^d$. With the notion available, several classical results
of the Euclidean setting became accessible even in general LCA groups. Here we
work out various versions in a general LCA group $G$ of the classical statement
that if a set $S\subset G$ has positive asymptotic uniform upper density, then
the difference set $S-S$ is syndetic.
