An exceptional set estimate for restricted projections to lines 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, $\det (\gamma (\theta ),{\gamma }^{\prime }(\theta ),{\gamma }^{\prime \prime }(\theta ))\ne 0$. For each $\theta \in [0,1]$, let $l_{\theta }=\{t\gamma (\theta ):t\in \mathbb{R}\}$ and ${\rho }_{\theta }:{\mathbb{R}}^3\to l_{\theta }$ be the orthogonal projections. We prove an exceptional set estimate. For any Borel set $A\subset {\mathbb{R}}^3$ and $0\le s\le 1$, define $E_s(A):=\{\theta \in [0,1]:\text{dim}({\rho }_{\theta }(A))<s\}$. We have $\text{dim}(E_s(A))\le 1+\frac{s-\text{dim}(A)}{2}$.

11 pages. arXiv admin note: substantial text overlap with arXiv:2207.13844