  We study the upper and lower regularity dimensions in relation to the notions
of doubling and uniformly perfect. These two regularity properties are closely
related which is quantified thanks to the regularity dimensions. The regularity
dimensions of pushforward measures onto graphs of Brownian motion are
calculated, similarly for pushforwards with respect to quasisymmetric
homeomorphisms. We finish by introducing an application to Diophantine
approximation in the setting of Kleinian groups.
