Assouad's theorem with dimension independent of the snowflaking

Assaf Naor; Ofer Neiman

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arXiv · 1012.2307

Assouad's theorem with dimension independent of the snowflaking

Assaf Naor·Ofer Neiman

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Abstract

It is shown that for every $K>0$ and $\e\in (0,1/2)$ there exist $N=N(K)\in \N$ and $D=D(K,\e)\in (1,\infty)$ with the following properties. For every separable metric space $(X,d)$ with doubling constant at most $K$, the metric space $(X,d^{1-\e})$ admits a bi-Lipschitz embedding into $\R^N$ with distortion at most $D$. The classical Assouad embedding theorem makes the same assertion, but with $N\to \infty$ as $\e\to 0$.

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Assaf Naor, Ofer Neiman. 2010-12-10. Assouad's theorem with dimension independent of the snowflaking. https://arxiv.org/abs/1012.2307

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