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

TL;DR

This paper constructs specific self-similar sets in Euclidean space with prescribed Hausdorff dimension and measure, answering an open question about their measure properties.

Contribution

It demonstrates the existence of self-similar sets with any Hausdorff dimension and measure ratio, advancing understanding of fractal geometry in Euclidean spaces.

Findings

Existence of self-similar sets with prescribed Hausdorff measure ratios

Construction method for such sets in $ eal^d$

Resolution of an open question by Zhiying Wen

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].