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Dvoretzky-type theorem for Ahlfors regular spaces

Volume 268 / 2023

Manor Mendel Studia Mathematica 268 (2023), 1-22 MSC: Primary 51F30; Secondary 28A78. DOI: 10.4064/sm210629-2-2 Published online: 22 July 2022

Abstract

It is proved that for any $0 \lt \beta \lt \alpha $, any bounded Ahlfors $\alpha $-regular space contains a $\beta $-regular compact subset that embeds biLipschitzly in an ultrametric with distortion at most $O(\alpha /(\alpha -\beta ))$. The bound on the distortion is asymptotically tight when $\beta \to \alpha $. The main tool used in the proof is a regular form of the ultrametric skeleton theorem.

Authors

  • Manor MendelDepartment of Mathematics and Computer Science
    The Open University of Israel
    1 University Road
    Raanana 43107, Israel
    e-mail

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