Ultrametric subsets with large Hausdorff dimension

Details Ultrametric subsets with large Hausdorff dimension Ultrametric subsets with large Hausdorff dimension

2011-06-05

arxiv.org/abs/1106.0879v3

Mathematics

Metric Geometry

the order of the lemmas has been changed, added figures

Scientific paper

It is shown that for every $\e\in (0,1)$, every compact metric space $(X,d)$ has a compact subset $S\subseteq X$ that embeds into an ultrametric space with distortion $O(1/\e)$, and $$\dim_H(S)\ge (1-\e)\dim_H(X),$$ where $\dim_H(\cdot)$ denotes Hausdorff dimension. The above $O(1/\e)$ distortion estimate is shown to be sharp via a construction based on sequences of expander graphs.

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