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Ultrametric subsets with large Hausdorff dimension

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Abstract

It is shown that for every ε∈(0,1), every compact metric space (X,d) has a compact subset S⊆X that embeds into an ultrametric space with distortion O(1/ε), and

$$\dim_H(S)\geqslant (1-\varepsilon)\dim_H(X),$$

where dim H (⋅) denotes Hausdorff dimension. The above O(1/ε) distortion estimate is shown to be sharp via a construction based on sequences of expander graphs.

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Authors and Affiliations

  1. Mathematics and Computer Science Department, Open University of Israel, 1 University Road, P.O. Box 808, Raanana, 43107, Israel

    Manor Mendel

  2. Courant Institute, New York University, 251 Mercer Street, New York, NY, 10012, USA

    Assaf Naor

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  1. Manor Mendel
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  2. Assaf Naor
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Correspondence to Assaf Naor.

Additional information

M.M. was partially supported by ISF grants 221/07 and 93/11, BSF grants 2006009 and 2010021, and a gift from Cisco Research Center. A.N. was partially supported by NSF grant CCF-0832795, BSF grants 2006009 and 2010021, and the Packard Foundation. Part of this work was completed when M.M. was visiting Microsoft Research and University of Washington, and A.N. was visiting the Discrete Analysis program at the Isaac Newton Institute for Mathematical Sciences and the Quantitative Geometry program at the Mathematical Sciences Research Institute.

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Mendel, M., Naor, A. Ultrametric subsets with large Hausdorff dimension. Invent. math. 192, 1–54 (2013). https://doi.org/10.1007/s00222-012-0402-7

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