法瓦尔德长度和定量可纠正性

arXiv - MATH - Metric Geometry Pub Date : 2024-08-07 DOI:arxiv-2408.03919

Damian Dąbrowski

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引用次数: 0

摘要

博尔集合 $E\subset\mathbb{R}^2$ 的法瓦尔德长度是其正交投影的平均长度。我们证明，如果 $E$ 是 Ahlfors 1-regular 且具有较大的 Favard 长度，那么它就包含了一大块 Lipschitz 图，这给出了贝西科维奇投影定理的定量版本。作为必然结果，我们回答了大卫和塞姆斯以及佩雷斯和索洛米亚克的问题。

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Favard length and quantitative rectifiability

The Favard length of a Borel set $E\subset\mathbb{R}^2$ is the average length of its orthogonal projections. We prove that if $E$ is Ahlfors 1-regular and it has large Favard length, then it contains a big piece of a Lipschitz graph. This gives a quantitative version of the Besicovitch projection theorem. As a corollary, we answer questions of David and Semmes and of Peres and Solomyak. We also make progress on Vitushkin's conjecture.

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arXiv - MATH - Metric Geometry

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