  We investigate the relationship between the dynamical properties of minimal
topological dynamical systems and the multiplicative combinatorial properties
of return time sets arising from those systems. In particular, we prove that
for a residual sets of points in any minimal system, the set of return times to
any non-empty, open set contains arbitrarily long geometric progressions. Under
the separate assumptions of total minimality and distality, we prove that
return time sets have positive multiplicative upper Banach density along
$\mathbb{N}$ and along multiplicative subsemigroups of $\mathbb{N}$,
respectively. The primary motivation for this work is the long-standing open
question of whether or not syndetic subsets of the positive integers contain
arbitrarily long geometric progressions; our main result is some evidence for
an affirmative answer to this question.
