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.
Original language | English (US) |
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Pages (from-to) | 1-54 |
Number of pages | 54 |
Journal | Inventiones Mathematicae |
Volume | 192 |
Issue number | 1 |
DOIs | |
State | Published - Jan 1 2013 |
Externally published | Yes |
All Science Journal Classification (ASJC) codes
- Mathematics(all)