Year: 2026
Type: article
Abstract: : Let $\gamma:[0,1]\rightarrow\mathbb{S}^2$ be a non-degenerate curve in $\mathbb{R}^3$, that is to say, $\det(\gamma(\theta),\gamma'(\theta),\linebreak\gamma" (\theta))\neq 0$. For each $\theta\in[0,1]$, let $V_\theta=\gamma(\theta)^\perp$ and let $\pi_\theta:\mathbb{R}^3\rightarrow V_\theta$ be the orthogonal projections. We prove that if $A\subset\mathbb{R}^3$ is a Borel set, then for a.e. $\theta\in [0,1]$ we have $\dim(\pi_\theta(A))=\min\{2,\dim A\}$. More generally, we prove an exceptional set estimate. For $A\subset\mathbb{R}^3$ and $0\le s\le 2$, define $E_s(A):=\{\theta\in[0,1]:\dim(\pi_\theta(A))<s\}$. We have $\dim(E_s(A))\le\max\{1+s-\dim(A),0\}$. We also prove that if $\dim(A)>2$, then for a.e. $\theta\in[0,1]$ we have $\mathcal{H}^2(\pi_\theta(A))>0$.
Source: American Journal of Mathematics
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Funder: National Science Foundation
OpenAlex ID: W4288804820