  It is well known and not difficult to prove that if $C$ of integers has
positive upper Banach density, the set of differences $C-C$ is syndetic, i.e.
the length of gaps is uniformly bounded. More surprisingly, Renling Jin showed
that whenever $A$ and $B$ have positive upper Banach density, then $A-B$ is
piecewise syndetic.
  Jin's result follows trivially from the first statement provided that $B$ has
large intersection with a shifted copy $A-n$ of $A$. Of course this will not
happen in general if we consider shifts by integers, but the idea can be put to
work if we allow "shifts by ultrafilters". As a consequence we obtain Jin's
Theorem.
