Accessible values of Assouad and the lower dimensions of subsets

Abstract

Let $E$ be a subset of a doubling metric space $(X,d)$. We prove that for any $s\in [0,{\dim }_{A}E]$, where ${\dim }_A$ denotes the Assouad dimension, there exists a subset $F$ of $E$ such that ${\dim }_{A}F=s$. We also show that the same statement holds for the lower dimension ${\dim }_L$.

13 pages

References in corpus (1)

• Weak separation condition, Assouad dimension, and Furstenberg homogeneity