Wild examples of rectifiable sets

TL;DR

This paper constructs two examples of rectifiable sets in the plane with finite length where the Jones function is nowhere locally integrable, illustrating diverse regularity properties and extending to higher dimensions.

Contribution

It provides explicit examples of rectifiable sets with pathological Jones function behavior, highlighting the complexity of geometric measure theory.

Findings

One set is connected, the other is Ahlfors regular.

Both sets have positive finite length.

Examples extend to higher dimensions and co-dimensions.

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 $\mathbb{R}^{2}$ 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.