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) |
|---|---|
| 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)
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Cite this
Mendel, M., & Naor, A. (2013). Ultrametric subsets with large Hausdorff dimension. Inventiones Mathematicae, 192(1), 1-54. https://doi.org/10.1007/s00222-012-0402-7