Geodesic
Wild examples of countably rectifiable sets
Max Goering1 ; Sean McCurdy2
1 University of Washington, Department of Mathematics
2 Carnegie Mellon University, Department of Mathematics
Annales Fennici Mathematici, Tome 46 (2021) no. 1, pp. 553-570

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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.
Keywords: Jones square function, rectifiability, traveling salesman, beta numbers

Max Goering 1 ; Sean McCurdy 2

1 University of Washington, Department of Mathematics
2 Carnegie Mellon University, Department of Mathematics 
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Max Goering; Sean McCurdy. Wild examples of countably rectifiable sets. Annales Fennici Mathematici, Tome 46 (2021) no. 1, pp. 553-570. http://geodesic.mathdoc.fr/item/AFM_2021_46_1_a32/