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

Computer Science discipline

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

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
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

Bibliographical note

Funding 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.

Access to Document

10.1007/s00222-012-0402-7

Cite this

@article{2f13a3e6719341c5853bda98ecda1249,

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",

note = "Funding 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.",

year = "2013",

month = apr,

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

language = "אנגלית",

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

N1 - Funding 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.

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

U2 - 10.1007/s00222-012-0402-7

DO - 10.1007/s00222-012-0402-7

M3 - ???researchoutput.researchoutputtypes.contributiontojournal.article???

AN - SCOPUS:84874956329

SN - 0020-9910

VL - 192

SP - 1

EP - 54

JO - Inventiones Mathematicae

JF - Inventiones Mathematicae

IS - 1

ER -

Ultrametric subsets with large Hausdorff dimension

Abstract

Bibliographical note

Access to Document

Other files and links

Fingerprint

Cite this