Ultrametric subsets with large Hausdorff dimension

Abstract

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.