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arXiv · 1905.01763

Wild examples of rectifiable sets

Max Goering·Sean McCurdy
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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.

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Max Goering, Sean McCurdy. 2019-05-05. Wild examples of rectifiable sets. https://arxiv.org/abs/1905.01763

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