Hausdorff dimension of wiggly metric spaces

Abstract

For a compact connected set $X\subseteq {\ell }^{\infty }$, we define a quantity ${\beta }^{\prime }(x,r)$ 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 ${\beta }^{\prime }(x,r)>\beta >0$ 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.

Corrected several errors and typos, improved exposition

References in corpus (1)

• Spaces and groups with conformal dimension greater than one

Cited by in corpus (2)

• Analytic capacity and dimension of sets with plenty of big projections
• Quasi-self-similar fractals containing "Y" have dimension larger than one