Abstract
We provide a sufficient Dini-type condition for a subset of a complete, quasiconvex metric space to be covered by a Hölder curve. This implies in particular that if the upper box-counting dimension is less than \(d \ge 1\), then it can be covered by an \(\frac{1}{d}\)-Hölder curve. On the other hand, for each \(1\le d <2\) we give an example of a compact set in the plane with lower box-counting dimension equal to zero and upper box-counting dimension equal to d, just failing the above Dini-type condition, that can not be covered by a countable collection of \(\frac{1}{d}\)-Hölder curves.
This is a preview of subscription content, log in via an institution to check access.
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
References
Azzam, J.: Hausdorff dimension of wiggly metric spaces. Ark. Mat. 53(1), 1–36 (2015)
Badger, M., Naples, L., Vellis, V.: Hölder curves and parametrizations in the analyst’s traveling salesman theorem. Adv. Math. 349, 564–647 (2019)
Badger, M., Vellis, V.: Geometry of measures in real dimensions via Hölder parameterizations. J. Geom. Anal. 29(2), 1153–1192 (2019)
Bishop, C.J., Jones, P.W.: Wiggly sets and limit sets. Ark. Mat. 35(2), 201–224 (1997)
Bishop, C.J., Peres, Y.: Fractals in Probability and Analysis. Cambridge Studies in Advanced Mathematics, vol. 162. Cambridge University Press, Cambridge (2017)
Burago, D., Burago, Y., Ivanov, S.: A Course in Metric Geometry. Graduate Studies in Mathematics, 33. AMS, Providence, RI (2001)
Falconer, K.: Fractal Geometry. Mathematical Foundations and Applications, 3rd edn. Wiley, Chichester (2014)
Federer, H.: Geometric Measure Theory. Die Grundlehren der mathematischen Wissenschaften, vol. 153. Springer, New York (1969)
Jones, P.W.: Rectifiable sets and the traveling salesman problem. Invent. Math. 102, 1–15 (1990)
Konyagin, S.V.: On the level sets of Lipschitz functions. Tatra Mt. Math. Publ. 2, 51–59 (1993)
Okikiolu, K.: Characterization of subsets of rectifiable curves in \({\mathbb{R}}^n\). J. Lond. Math. Soc. 46(2), 336–348 (1992)
Schul, R.: Subsets of rectifiable curves in Hilbert space—the analyst’s TSP. J. Anal. Math. 103, 331–375 (2007)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This research was partially supported by the Swiss National Science Foundation.
Rights and permissions
About this article
Cite this article
Balogh, Z.M., Züst, R. Box-counting by Hölder’s traveling salesman. Arch. Math. 114, 561–572 (2020). https://doi.org/10.1007/s00013-019-01415-5
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00013-019-01415-5