正则超度量骨架
Regular Ultrametric Skeletons
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中文总结 AI 辅助
研究任意紧致度量空间，基于Bartal的拉姆齐分解等，证明具有线性扩张控制球的正则版本的超度量骨架定理。
中文摘要 AI 辅助
Mendel和Naor（2013年）的定理将每个紧致度量概率空间与一个大的超度量子集相关联，该子集带有一个概率测度，其球上的测度由原始测度从上方控制。基于Bartal的拉姆齐分解（2021年）以及作者之前关于加倍空间的超度量骨架定理的证明（2022年、2023年），我们证明了具有线性扩张控制球的任意紧致度量空间的正则版本。
英文摘要
The ultrametric skeleton theorem extracts from every compact metric probability space a subset of ultrametric distortion $O(1/\varepsilon)$ that carries a measure whose balls are controlled by the $(1-\varepsilon)$-power of the original measure on dilated concentric balls. We prove a two-sided version for arbitrary compact metric spaces: for every ball centered on the skeleton, the skeleton measure also has a lower bound in terms of the original measure on a smaller nonconcentric ball contained in it. We also give a short proof of the original skeleton theorem and improve the dilation of its control balls from $\exp(O(1/\varepsilon^2))$ to $O(1/\varepsilon)$.
