Geometric and Functional Analysis ( IF 2.4 ) Pub Date : 2024-02-05 , DOI: 10.1007/s00039-024-00660-3 Tuomas Orponen , Pablo Shmerkin , Hong Wang
We provide several new answers on the question: how do radial projections distort the dimension of planar sets? Let \(X,Y \subset \mathbb{R}^{2}\) be non-empty Borel sets. If X is not contained in any line, we prove that
$$ \sup _{x \in X} \dim _{\mathrm {H}}\pi _{x}(Y \, \setminus \, \{x\}) \geq \min \{ \dim _{\mathrm {H}}X,\dim _{\mathrm {H}}Y,1\}. $$If dimHY>1, we have the following improved lower bound:
$$ \sup _{x \in X} \dim _{\mathrm {H}}\pi _{x}(Y \, \setminus \, \{x\}) \geq \min \{ \dim _{\mathrm {H}}X + \dim _{\mathrm {H}}Y - 1,1\}. $$Our results solve conjectures of Lund-Thang-Huong, Liu, and the first author. Another corollary is the following continuum version of Beck’s theorem in combinatorial geometry: if \(X \subset \mathbb{R}^{2}\) is a Borel set with the property that dimH(X ∖ ℓ)=dimHX for all lines \(\ell \subset \mathbb{R}^{2}\), then the line set spanned by X has Hausdorff dimension at least min{2dimHX,2}.
While the results above concern \(\mathbb{R}^{2}\), we also derive some counterparts in \(\mathbb{R}^{d}\) by means of integralgeometric considerations. The proofs are based on an ϵ-improvement in the Furstenberg set problem, due to the two first authors, a bootstrapping scheme introduced by the second and third author, and a new planar incidence estimate due to Fu and Ren.
中文翻译：
径向投影的考夫曼和福尔科纳估计以及贝克定理的连续版
我们对这个问题提供了几个新的答案：径向投影如何扭曲平面集合的维度？设\(X,Y \subset \mathbb{R}^{2}\)为非空 Borel 集。如果X不包含在任何行中，我们证明
$$ \sup _{x \in X} \dim _{\mathrm {H}}\pi _{x}(Y \, \setminus \, \{x\}) \geq \min \{ \dim _ {\mathrm {H}}X，\dim _{\mathrm {H}}Y,1\}。 $$如果暗淡H Y >1，我们有以下改进的下界：
$$ \sup _{x \in X} \dim _{\mathrm {H}}\pi _{x}(Y \, \setminus \, \{x\}) \geq \min \{ \dim _ {\mathrm {H}}X + \dim _{\mathrm {H}}Y - 1,1\}。 $$我们的结果解决了 Lund-Thang-Huong、Liu 和第一作者的猜想。另一个推论是组合几何中贝克定理的以下连续统版本：如果\(X \subset \mathbb{R}^{2}\)是一个 Borel 集，其属性为 dim H ( X ∖ ℓ )=dim H X对于所有线\(\ell \subset \mathbb{R}^{2}\) ，则X跨越的线集的豪斯多夫维度至少为 min{2dim H X ,2}。
虽然上面的结果涉及\(\mathbb{R}^{2}\)，但我们还通过积分几何考虑推导出\(\mathbb{R}^{d}\)中的一些对应项。这些证明基于Furstenberg 集问题中的ϵ改进（归功于两位第一作者）、第二位作者和第三位作者引入的引导方案，以及归功于 Fu 和 Ren 的新平面发生率估计。
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