Ultrametric subsets with large Hausdorff dimension
Seminar Room 1, Newton Institute
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.