Ultrametric subsets with large Hausdorff dimension

Manor Mendel, Assaf Naor

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.

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2.3 years ago

Omar ▴ 30

In the construction of the expander fractals was there anything special about using the Erdos-Renyi model for random graphs or could you have done the same or similar construction using other random models for graphs?

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by Omar ▴ 30

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Ultrametric subsets with large Hausdorff dimension