Kahane's upper density and syndetic sets in LCA groups

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.