Ultrametric subsets with large Hausdorff dimension

Manor Mendel; Assaf Naor

doi:10.1007/s00222-012-0402-7

Ultrametric subsets with large Hausdorff dimension

Manor Mendel, Assaf Naor

Research output: Contribution to journal › Article › peer-review

10 Scopus citations

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),≥(1-ε)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.

Original language	English (US)
Pages (from-to)	1-54
Number of pages	54
Journal	Inventiones Mathematicae
Volume	192
Issue number	1
DOIs	https://doi.org/10.1007/s00222-012-0402-7
State	Published - Apr 2013
Externally published	Yes

All Science Journal Classification (ASJC) codes

Mathematics(all)

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AB - 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 dimH(S),≥(1-ε)dimH(X) where dimH(·) 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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Ultrametric subsets with large Hausdorff dimension

Abstract

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