Ultrametric subsets with large Hausdorff dimension

TL;DR

The paper proves that any compact metric space contains a large Hausdorff dimension subset that can be embedded into an ultrametric space with distortion proportional to 1/epsilon, and this bound is shown to be optimal.

Contribution

It establishes the existence of large Hausdorff dimension subsets with near-isometric ultrametric embeddings and proves the optimality of the distortion bound.

Findings

Existence of subsets with Hausdorff dimension at least (1-e) times the original.

Ultrametric embedding distortion is bounded by O(1/epsilon).

The distortion bound is proven to be sharp using expander graph constructions.

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.