Hausdorff dimension of wiggly metric spaces

TL;DR

This paper establishes a lower bound on the Hausdorff dimension of certain compact connected sets in ll^{} based on their local approximability by geodesic curves, generalizing previous results and answering open questions.

Contribution

It introduces a new quantitative measure eta' that assesses local geodesic approximation and proves a dimension lower bound related to this measure, extending prior theorems.

Findings

Sets with uniformly bounded eta' have Hausdorff dimension greater than 1 plus a quadratic function of eta'

The result generalizes a theorem of Bishop and Jones

Answers a question posed by Bishop and Tyson

Abstract

For a compact connected set $X\subseteq \ell^{\infty}$, we define a quantity $\beta'(x,r)$ that measures how close $X$ may be approximated in a ball $B(x,r)$ by a geodesic curve. We then show there is $c>0$ so that if $\beta'(x,r)>\beta>0$ for all $x\in X$ and $r<r_{0}$, then $\dim X>1+c\beta^{2}$. This generalizes a theorem of Bishop and Jones and answers a question posed by Bishop and Tyson.