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<r 0, 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.
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The author was supported by the NSF grants RTG DMS 08-38212 and DMS-0856687.
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Azzam, J. Hausdorff dimension of wiggly metric spaces. Ark Mat 53, 1–36 (2015). https://doi.org/10.1007/s11512-014-0197-4
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DOI: https://doi.org/10.1007/s11512-014-0197-4