平面中不可求长集合的定量可见性估计
Quantitative visibility estimates for unrectifiable sets in the plane
AI总结：
本文定量研究平面集合的可见性，给出不可求长自相似一维集合δ-邻域可见性的上界，并借助Bourgain离散化和积估计证明更一般集合δ-邻域可见性的下界。
摘要：
平面集合 $S$ 从点 $a$ 的“可见性”定义为 $S$ 从 $a$ 到以 $a$ 为圆心的单位圆的径向投影的归一化大小。Simon 和 Solomyak（Real Anal. Exchange 2006/07）证明了不可求长的自相似一维集合从平面中每一点看都是不可见的。我们通过给出这类集合的 $δ$-邻域可见性的上界来对此进行量化。我们还基于 Bourgain 的离散化和积估计，证明了更一般集合的 $δ$-邻域可见性的下界。
英文摘要：
The "visibility" of a planar set $S$ from a point $a$ is defined as the normalized size of the radial projection of $S$ from $a$ to the unit circle centered at $a$. Simon and Solomyak (Real Anal. Exchange 2006/07) proved that unrectifiable self-similar one-sets are invisible from every point in the plane. We quantify this by giving an upper bound on the visibility of $δ$-neighbourhoods of such sets. We also prove lower bounds on the visibility of $δ$-neighborhoods of more general sets, based in part on Bourgain's discretized sum-product estimates
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
