Hausdorff dimension of wiggly metric spaces

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