On the upper regularity dimensions of measures

We study the upper regularity dimension which describes the extremal local scaling behaviour of a measure and effectively quantifies the notion of doubling. We conduct a thorough study of the upper regularity dimension, including its relationship with other concepts such as the Assouad dimension, the upper local dimension, the Lq-spectrum and weak tangent measures. We also compute the upper regularity dimension explicitly in a number of important contexts including self-similar measures, self-affine measures, and measures on sequences