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.
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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 -