Wild examples of countably rectifiable sets

Abstract

We study the geometry of sets based on the behavior of the Jones function, $J_E(x)={\int }_0^1{\beta }_{E;2}^1(x,r{)}^2\frac{dr}{r}$. We construct two examples of countably 1-rectifiable sets in ${\mathbf{R}}^2$ with positive and finite ${\mathcal{H}}^1$-measure for which the Jones function is nowhere locally integrable. These examples satisfy different regularity properties: one is connected and one is Ahlfors regular. Both examples can be generalized to higher-dimension and co-dimension.