On Hausdorff dimension of radial projections

TL;DR

This paper investigates the Hausdorff dimension preservation under radial projections for sets in Euclidean space, providing bounds on the size of the set of points where the dimension drops, and improves existing results in certain cases.

Contribution

The authors develop a framework linking radial projections of different sets and improve bounds on the dimension of points causing dimension drop, extending previous results.

Findings

Bound on the Hausdorff dimension of points where dimension drops

Improved bounds for sets with dimension in (d-2, d-1]

Optimal result at the endpoint dimension d-1

Abstract

For any $x\in\mathbb{R}^d$, $d\geq 2$, denote $\pi^x: \mathbb{R}^d\backslash\{x\}\rightarrow S^{d-1}$ as the radial projection $$\pi^x(y)=\frac{y-x}{|y-x|}. $$ Given a Borel set $E\subset{\Bbb R}^d$, $\dim_{\mathcal{H}} E\leq d-1$, 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 $\dim_{\mathcal{H}}\pi^x(E)=\dim_{\mathcal{H}} E$. 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, whether $\dim_{\mathcal{H}}\pi^x(E)=\dim_{\mathcal{H}}E$ for some $x\in F$ can be reduced to whether $F$ is visible from some $y\in E$ (i.e. $\mathcal{H}^{d-1}(\pi^y(F))>0$). This allows us to apply Orponen's estimate on visibility to obtain $$\dim_{\mathcal{H}}\left\{x\in\mathbb{R}^d: \dim_{\mathcal{H}}\pi^x(E)<\dim_{\mathcal{H}}E\right\}\leq 2(d-1)-\dim_{\mathcal{H}}E,$$ for any Borel set $E\subset{\Bbb R}^d$, $\dim_{\mathcal{H}} E\in(d-2, d-1]$. This improves the Peres-Schlag bound when $\dim_{\mathcal{H}} E\in(d-\frac{3}{2}, d-1]$, and it is optimal at the endpoint $\dim_{\mathcal{H}} E=d-1$.