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.