Abstract
For a compact connected set X⊆ℓ∞, we define a quantity β′(x, r) that measures how close X may be approximated in a ball B(x, r) by a geodesic curve. We then show that there is c>0 so that if β′(x, r)> β>0 for all x∈X and r< r0, then $\operatorname{dim}X>1+c\beta^{2}$. This generalizes a theorem of Bishop and Jones and answers a question posed by Bishop and Tyson.
Funding Statement
The author was supported by the NSF grants RTG DMS 08-38212 and DMS-0856687.
Citation
Jonas Azzam. "Hausdorff dimension of wiggly metric spaces." Ark. Mat. 53 (1) 1 - 36, April 2015. https://doi.org/10.1007/s11512-014-0197-4
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