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

13 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)

Access to Document

10.1007/s00222-012-0402-7

Cite this

@article{ff9c742dc2d1420ab9acb2c8f35326e8,

title = "Ultrametric subsets with large Hausdorff dimension",

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 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.",

author = "Manor Mendel and Assaf Naor",

year = "2013",

month = apr,

doi = "10.1007/s00222-012-0402-7",

language = "English (US)",

volume = "192",

pages = "1--54",

journal = "Inventiones Mathematicae",

issn = "0020-9910",

publisher = "Springer New York",

number = "1",

}

TY - JOUR

T1 - Ultrametric subsets with large Hausdorff dimension

AU - Mendel, Manor

AU - Naor, Assaf

PY - 2013/4

Y1 - 2013/4

N2 - 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.

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.

UR - http://www.scopus.com/inward/record.url?scp=84874956329&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84874956329&partnerID=8YFLogxK

U2 - 10.1007/s00222-012-0402-7

DO - 10.1007/s00222-012-0402-7

M3 - Article

AN - SCOPUS:84874956329

VL - 192

SP - 1

EP - 54

JO - Inventiones Mathematicae

JF - Inventiones Mathematicae

SN - 0020-9910

IS - 1

ER -

Ultrametric subsets with large Hausdorff dimension

Abstract

All Science Journal Classification (ASJC) codes

Access to Document

Other files and links

Fingerprint

Cite this