论文标题
在一组径向预测中
On exceptional sets of radial projections
论文作者
论文摘要
我们证明了飞机中径向投影的两个新的异常集估计。如果$ k \ subset \ mathbb {r}^{2} $是borel设置，带有$ \ dim _ {\ mathrm {h}} k> 1 $，然后$ \ dim _ {\ dim _ {\ mathRm { \ dim _ {\ mathrm {h}}π_{x}（k）\ leqσ\} \ leq \ max \ max \ {1 +σ-\ \ dim _ {\ mathrm {h} \ Mathbb {r}^{2} $是一个borel设置，带有$ \ dim _ {\ mathrm {h}} k \ leq 1 $，然后$ \ \ dim _ {\ dim _ {\ mathrm {h}}} \ in \ in \ in \ in \ in \ in \ in \ m \ dim _ {\ mathrm {h}}π_{x}（k）<\ dim _ {\ mathrm {\ mathrm {h}} k \} \ leq 1。我们的结果解决了隆德 - 洪周四和刘的猜想的平面案例。
We prove two new exceptional set estimates for radial projections in the plane. If $K \subset \mathbb{R}^{2}$ is a Borel set with $\dim_{\mathrm{H}} K > 1$, then $$\dim_{\mathrm{H}} \{x \in \mathbb{R}^{2} \, \setminus \, K : \dim_{\mathrm{H}} π_{x}(K) \leq σ\} \leq \max\{1 + σ- \dim_{\mathrm{H}} K,0\}, \qquad σ\in [0,1).$$ If $K \subset \mathbb{R}^{2}$ is a Borel set with $\dim_{\mathrm{H}} K \leq 1$, then $$\dim_{\mathrm{H}} \{x \in \mathbb{R}^{2} \, \setminus \, K : \dim_{\mathrm{H}} π_{x}(K) < \dim_{\mathrm{H}} K\} \leq 1.$$ The finite field counterparts of both results above were recently proven by Lund, Thang, and Huong Thu. Our results resolve the planar cases of conjectures of Lund-Thang-Huong Thu, and Liu.
