An exceptional set estimate for restricted projections to lines in $\mathbb{R}^3$

Abstract

Let $γ:[0,1]\rightarrow \mathbb{S}^{2}$ be a non-degenerate curve in $\mathbb{R}^3$, that is to say, $\det\big(γ(θ),γ'(θ),γ''(θ)\big)\neq 0$. For each $θ\in[0,1]$, let $l_θ=\{tγ(θ):t\in\mathbb{R}\}$ and $ρ_θ:\mathbb{R}^3\rightarrow l_θ$ 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 $E_s(A):=\{θ\in[0,1]: \text{dim}(ρ_θ(A))<s\}$. We have $\text{dim}(E_s(A))\le 1+\frac{s-\text{dim}(A)}{2}$.