A note on Hausdorff measures of self-similar sets in ${\mathbb{R}}^d$

Abstract

We prove that for all $s\in (0,d)$ and $c\in (0,1)$ there exists a self-similar set $E\subset {\mathbb{R}}^d$ with Hausdorff dimension $s$ such that ${\mathcal{H}}^s(E)=c|E{|}^s$. This answers a question raised by Zhiying Wen[16].