论文标题
Kaufman和Falconer估计径向预测和贝克定理的连续版本
Kaufman and Falconer estimates for radial projections and a continuum version of Beck's Theorem
论文作者
论文摘要
我们在这个问题上提供了几个新答案：径向投影如何扭曲平面集的维度？令$ x，y \ subset \ mathbb {r}^{2} $为非空的borel集。如果$ x $不包含在任何行上，我们证明\ [\ sup_ {x \ in x} \ dim _ {\ mathrm {\ mathrm {h}}π_{x}（x}（x}（y）\ geq \ geq \ min \ min \ min \ min \ {\ dim _ {\ dim _ { Y，1 \}。 \]如果$ \ dim _ {\ mathrm {h}} y> 1 $，我们具有以下改进的下限：\ [\ [\ sup_ {x \ in x}} \ dim _ {\ dim _ {\ mathrm {h}}}π_ \ min \ {\ dim _ {\ mathrm {h}} x + \ dim _ {\ mathrm {h}} y -1,1 \}。 \]我们的结果解决了隆德，刘，第一作者的猜想。另一个推论是贝克（Beck）定理的以下连续版本：如果$ x \ subset \ mathbb {r}^{2} $是borel设置，其属性具有$ \ dim _ {\ mathrm {h}}}}（ \ dim _ {\ mathrm {h}} x $用于所有行$ \ ell \ ell \ subset \ subset \ mathbb {r}^{2} $，然后由$ x $跨越的行设置为hausdorff dimension，至少$ \ min \ min \ min \ {2 \ dim _ dim _ {\ dim _ {\ mathrm {\ mathrm { 虽然上面的结果涉及$ \ mathbb {r}^{2} $，但我们还通过整合地位考虑因素来得出$ \ mathbb {r}^{d} $中的一些对应物。这些证明是基于Furstenberg Set问题中的$ε$改进，这是由于两位第一作者，第二和第三作者引入的引导程序方案以及由于FU和REN引起的新的平面发病率估算。
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 on any line, we prove that \[ \sup_{x \in X} \dim_{\mathrm{H}} π_{x}(Y) \geq \min\{\dim_{\mathrm{H}} X,\dim_{\mathrm{H}} Y,1\}. \] If $\dim_{\mathrm{H}} Y > 1$, we have the following improved lower bound: \[ \sup_{x \in X} \dim_{\mathrm{H}} π_{x}(Y \, \setminus \, \{x\}) \geq \min\{\dim_{\mathrm{H}} X + \dim_{\mathrm{H}} Y - 1,1\}. \] 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 $\dim_{\mathrm{H}} (X \, \setminus \, \ell) = \dim_{\mathrm{H}} X$ for all lines $\ell \subset \mathbb{R}^{2}$, then the line set spanned by $X$ has Hausdorff dimension at least $\min\{2\dim_{\mathrm{H}} X,2\}$. 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.
