  We study the action of the multiplicative group generated by two prime
numbers in $\mathbf{Z}/Q\mathbf{Z}$. More specifically, we study returns to the
set $([-Q^\varepsilon,Q^\varepsilon]\cap \mathbf{Z})/Q\mathbf{Z}$. This is
intimately related to the problem of bounding the greatest common divisor of
$S$-unit differences, which we revisit. Our main tool is the $S$-adic subspace
theorem.
