On restricted projections to planes in ${\mathbb{R}}^3$

Abstract

Let $\gamma :[0,1]\to {\mathbb{S}}^2$ be a non-degenerate curve in ${\mathbb{R}}^3$, that is to say, $\det (\gamma (\theta ),{\gamma }^{\prime }(\theta ),\gamma "(\theta ))\ne 0$. For each $\theta \in [0,1]$, let $V_{\theta }=\gamma (\theta {)}^{\perp }$ and let ${\pi }_{\theta }:{\mathbb{R}}^3\to V_{\theta }$ 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 ${\mathcal{H}}^2({\pi }_{\theta }(A))>0$.

39 pages, 2 figures; accepted by Amer. J. Math

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