Inventiones mathematicae

, Volume 192, Issue 1, pp 1–54 | Cite as

Ultrametric subsets with large Hausdorff dimension

  • Manor Mendel
  • Assaf Naor


It is shown that for every ε∈(0,1), every compact metric space (X,d) has a compact subset SX 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.

Mathematics Subject Classification

30L05 46B85 37F35 


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Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  1. 1.Mathematics and Computer Science DepartmentOpen University of IsraelRaananaIsrael
  2. 2.Courant InstituteNew York UniversityNew YorkUSA

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