I will present two results joint with Lingmin Liao. The first is
calculation of Hausdorff dimension of the multiplicative golden shift and
related multiplicative multifractal sets in the setting when the maps are
linear but not with a constant slope. The second is solving a question about
inhomogeneous Diophantine approximations: how big is the set of points
belonging to infinitely many balls of form B(nα,rn)
depending on the Diophantine type of α. This strengthens results
of Bugeaud, Troubetskoy&Schmelling, and Xu.