Menger curvature and $C^{1}$ regularity of fractals

We show that if $E$ is an $s$-regular set in $\mathbf{R}^{2}$ for which the triple integral $\int _{E}\int _{E}\int _{E}c(x,y,z)^{2s}\,d\mathcal{H}^{s}x\,d\mathcal{H}^{s}y\,d \mathcal{H}^{s}z$of the Menger curvature $c$ is finite and if $0<s\le 1/2$, then $\mathcal{H}^{s}$almost all of $E$ can be covered with countably many $C^{1}$ curves. We give an example to show that this is false for $1/2<s<1$.