当前位置: X-MOL 学术Rev. Mat. Iberoam. › 论文详情
Our official English website, www.x-mol.net, welcomes your feedback! (Note: you will need to create a separate account there.)
On Hausdorff dimension of radial projections
Revista Matemática Iberoamericana ( IF 1.2 ) Pub Date : 2020-10-26 , DOI: 10.4171/rmi/1227
Bochen Liu 1
Affiliation  

For any $x\in\mathbb{R}^d$, $d\geq 2$, denote by $\pi^x\colon \mathbb{R}^d\backslash\{x\}\rightarrow S^{d-1}$ the radial projection $$ \pi^x(y)=\frac{y-x}{|y-x|}\cdot $$ Given a Borel set $E\subset\mathbb R^d$ with $\mathrm{dim}_{\mathcal{H}} E\leq d-1$, in this paper we investigate for how many $x\in \mathbb{R}^d$ the radial projection $\pi^x$ preserves the Hausdorff dimension of $E$, namely whether $\mathrm{dim}_{\mathcal{H}}\pi^x(E)=\mathrm{dim}_{\mathcal{H}} E$. We develop a general framework to link $\pi^x(E)$, $x\in F$, and $\pi^y(F)$, $y\in E$, for any Borel set $F\subset\mathbb{R}^d$. In particular, this allows us to apply Orponen's estimate on visibility to study whether $\mathrm {dim}_{\mathcal{H}}\pi^x(E)=\mathrm{dim}_{\mathcal{H}}E$ for some $x\in F$. More precisely, we show $$ \mathrm{dim}_{\mathcal{H}}\{x\in\mathbb{R}^d: \mathrm {dim}_{\mathcal{H}}\pi^x(E) < \mathrm {dim}_{\mathcal{H}}E\}\leq 2(d-1)-\mathrm {dim}_{\mathcal{H}}E, $$ for any Borel set $E\subset\mathbb R^d$ \n{with} $\mathrm{dim}_{\mathcal{H}} E\in(d-2, d-1]$. This improves the Peres–Schlag bound when $\mathrm{dim}_{\mathcal{H}} E\in(d-{3}/{2}, d-1]$, and it is optimal at the endpoint $\mathrm{dim}_{\mathcal{H}} E=d-1$.

中文翻译:

关于径向投影的豪斯多夫维数

对于任何 $x\in\mathbb{R}^d$, $d\geq 2$,表示为 $\pi^x\colon \mathbb{R}^d\backslash\{x\}\rightarrow S^{ d-1}$ 径向投影 $$ \pi^x(y)=\frac{yx}{|yx|}\cdot $$ 给定一个 Borel 集 $E\subset\mathbb R^d$ 和 $\mathrm {dim}_{\mathcal{H}} E\leq d-1$,在本文中我们研究了径向投影 $\pi^x$ 保留了多少 $x\in \mathbb{R}^d$ $E$的豪斯多夫维数,即是否$\mathrm{dim}_{\mathcal{H}}\pi^x(E)=\mathrm{dim}_{\mathcal{H}} E$。我们开发了一个通用框架来链接 $\pi^x(E)$, $x\in F$ 和 $\pi^y(F)$, $y\in E$,对于任何 Borel 集合 $F\subset \mathbb{R}^d$。特别是,这允许我们应用 Orponen 对可见性的估计来研究 $\mathrm {dim}_{\mathcal{H}}\pi^x(E)=\mathrm{dim}_{\mathcal{H}} E$ 为一些 $x\in F$。更准确地说,我们显示 $$ \mathrm{dim}_{\mathcal{H}}\{x\in\mathbb{R}^d:
更新日期:2020-10-26
down
wechat
bug