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Ultrametric subsets with large Hausdorff dimension

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If you have a question about this talk, please contact Mustapha Amrani.

Discrete Analysis

We show that for any 1>ε>0, any metric space X contains a subset Y which is O(1/ε) equivalent to an ultrametric and dimH(Y)>(1-ε)dimH(X), where dimH is the Hausdorff dimension. The dependence on ε is tight up-to a constant multiplicative factor.

This result can be viewed as high distortion metric analog of Dvoretzky theorem. Low distortion analog of Dvoretzky theorem is impossible since there are examples of compact metric spaces of arbitrary large Hausdorff dimension for which any subset that embeds in Hilbert space with distortion smaller than 2 must have zero Hausdorff dimension.

This talk is part of the Isaac Newton Institute Seminar Series series.

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