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

Kahane's upper density and syndetic sets in LCA groups