Measures with specified support and arbitrary Assouad dimensions

Abstract

We show that if the upper Assouad dimension of the compact set $E\subseteq \mathbb{R}$ is positive, then given any $D>{\dim }_{A}E$ there is a measure with support $E$ and upper Assouad (or regularity) dimension $D$. Similarly, given any $0\le d<{\dim }_{L}E,$ there is a measure on $E$ with lower Assouad dimension $d$.