标度无关的度量碎裂与非线性Dvoretzky定理
Scale-oblivious metric fragmentation and the nonlinear Dvoretzky theorem
摘要：
我们引入了一个随机迭代分裂过程有限度量空间，这是保证导致一个多项式的大子集，是$D$ -相当于超度量，其中$D\in (2,\infty)$是一个规定的目标失真。由于这一程序适用于任意接近非线性Dvoretzky相变失真2的$D$，因此我们得到了一个更简单的概率证明的主要结果的巴特尔，Linial，孟德尔，和Naor，回答了一个问题，从孟德尔和Naor，并产生最知名的非线性Dvoretzky定理的界限。我们的方法利用了一系列随机尺度，在给定的度量空间被分割。正如嵌入理论中许多先前的随机化论证一样，这些尺度的选择与所讨论的度量空间的几何无关。我们表明，我们的界限是尖锐的，如果利用这样一个“规模遗忘”的碎片化过程。
英文摘要：
We introduce a randomized iterative fragmentation procedure for finite metric spaces, which is guaranteed to result in a polynomially large subset that is $D$-equivalent to an ultrametric, where $D\in (2,\infty)$ is a prescribed target distortion. Since this procedure works for $D$ arbitrarily close to the nonlinear Dvoretzky phase transition at distortion 2, we thus obtain a much simpler probabilistic proof of the main result of Bartel, Linial, Mendel, and Naor, answering a question from Mendel and Naor, and yielding the best known bounds in the nonlinear Dvoretzky theorem. Our method utilizes a sequence of random scales at which a given metric space is fragmented. As in many previous randomized arguments in embedding theory, these scales are chosen irrespective of the geometry of the metric space in question. We show that our bounds are sharp if one utilizes such a "scale-oblivious" fragmentation procedure.
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