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A simple proof of Dvoretzky-type theorem for Hausdorff dimension in doubling spaces. (English) Zbl 1495.46016

One of the fundamental results of M. Mendel and A. Naor [Invent. Math. 192, No. 1, 1–54 (2013; Zbl 1272.30082)] is:
For every \(\varepsilon\in (0, 1)\), every compact metric space \((X,d)\) has a compact subset \(S\subseteq X\) that embeds into an ultrametric space with distortion \(O(1/\varepsilon)\), and the Hausdorff dimension \(\dim_H(S)\ge (1-\varepsilon) \dim_H(X)\).
This result can be derived from the result called the ultrametric skeleton theorem proved in [M. Mendel and A. Naor, Proc. Natl. Acad. Sci. USA 110, No. 48, 19256–19262 (2013; Zbl 1307.46013)]
A metric space \((X,d)\) is called \(\lambda\)-doubling if each bounded set \(Z\) in \(X\) can be covered by \(\lambda\) sets of diameter \(\mathrm{diam}(Z)/2\). The main goal of the paper under review is to find a simple proof of the ultrametric skeleton theorem in the special case of doubling spaces.
The obtained estimates for the important parameters are in some respects better than the estimates in [M. Mendel and A. Naor, Proc. Natl. Acad. Sci. USA 110, No. 48, 19256–19262 (2013; Zbl 1307.46013)].
In the proof, the author uses Y. Bartal’s Ramsey decomposition [“Advances in metric Ramsey theory and its applications”, Preprint (2021), arXiv:2104.03484]. The author answers an open problem from [O. Zindulka, Int. Math. Res. Not. 2020, No. 3, 698–721 (2020; Zbl 1435.28007)].

MSC:

46B85 Embeddings of discrete metric spaces into Banach spaces; applications in topology and computer science
51F30 Lipschitz and coarse geometry of metric spaces
28A78 Hausdorff and packing measures
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References:

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