Wild examples of countably rectifiable sets
Keywords:
Jones square function, rectifiability, traveling salesman, beta numbersAbstract
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.How to Cite
Goering, M., & McCurdy, S. (2021). Wild examples of countably rectifiable sets. Annales Fennici Mathematici, 46(1), 553–570. Retrieved from https://afm.journal.fi/article/view/109631
Copyright (c) 2021 The Finnish Mathematical Society
![Creative Commons License](http://i.creativecommons.org/l/by-nc/4.0/88x31.png)
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.