论文标题
定量besicovitch投影定理，用于不规则方向集
Quantitative Besicovitch projection theorem for irregular sets of directions
论文作者
论文摘要
经典的besicovitch投影定理指出，如果具有有限长度的平面设置$ e $纯粹是不可分割的，那么几乎所有$ e $的正交投影的长度为零。我们证明了此结果的定量版本：如果$ e \ subset \ mathbb {r}^2 $是ad-nigular的，并且存在一组方向$ g \ subset \ subset \ mathbb {s}^1 $ at $ \ mathcal {h}^h}^1（h}^1（g） $ \ |π_θ\ Mathcal {h}^1 | _e \ | _ {l^{\ infty}}} \ sillsim 1 $，那么Lipschitz Graph $γ$可以用$ \ mathrm {lip}（lip}（γγ）\ simsim 1 $ 1 $ 1 $ 1 $ 1 $ 1 $ 1 $。我们结果的主要新颖性是，$ g $的一组良好的方向仅是可测量的，并且在衡量标准方面，而先前的结果则需要$ g $作为弧线。作为推论，我们会在AD定型集中获得避免大量方向的结果，从某种意义上说，它们跨越的方向集具有较大的补充。它概括了以下简单的观察：一组$ e $包含在Lipschitz图中，并且仅当$ e $跨越的方向的补充包含弧线时。
The classical Besicovitch projection theorem states that if a planar set $E$ with finite length is purely unrectifiable, then almost all orthogonal projections of $E$ have zero length. We prove a quantitative version of this result: if $E\subset\mathbb{R}^2$ is AD-regular and there exists a set of direction $G\subset \mathbb{S}^1$ with $\mathcal{H}^1(G)\gtrsim 1$ such that for every $θ\in G$ we have $\|π_θ\mathcal{H}^1|_E\|_{L^{\infty}}\lesssim 1$, then a big piece of $E$ can be covered by a Lipschitz graph $Γ$ with $\mathrm{Lip}(Γ)\lesssim 1$. The main novelty of our result is that the set of good directions $G$ is assumed to be merely measurable and large in measure, while previous results of this kind required $G$ to be an arc. As a corollary, we obtain a result on AD-regular sets which avoid a large set of directions, in the sense that the set of directions they span has a large complement. It generalizes the following easy observation: a set $E$ is contained in some Lipschitz graph if and only if the complement of the set of directions spanned by $E$ contains an arc.
