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Mathematics > Classical Analysis and ODEs
Title: On restricted projections to planes in $\mathbb{R}^3$
(Submitted on 28 Jul 2022 (v1), last revised 24 Mar 2024 (this version, v2))
Abstract: Let $\gamma:[0,1]\rightarrow \mathbb{S}^{2}$ be a non-degenerate curve in $\mathbb{R}^3$, that is to say, $\det\big(\gamma(\theta),\gamma'(\theta),\gamma"(\theta)\big)\neq 0$. 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 $\text{dim}(\pi_\theta(A))=\min\{2,\text{dim} A\}$. More generally, we prove an exceptional set estimate. For $A\subset\mathbb{R}^3$ and $0\le s\le 2$, define $E_s(A):=\{\theta\in[0,1]: \text{dim}(\pi_\theta(A))<s\}$. We have $\text{dim}(E_s(A))\le 1+s-\text{dim}(A)$. We also prove that if $\text{dim}(A)>2$, then for a.e. $\theta\in[0,1]$ we have $\mathcal{H}^2(\pi_\theta (A))>0$.
Submission history
From: Shengwen Gan [view email][v1] Thu, 28 Jul 2022 01:06:37 GMT (38kb)
[v2] Sun, 24 Mar 2024 12:44:39 GMT (40kb)
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