论文标题
在$ \ mathbb {r}^3 $中对飞机的限制预测
On restricted projections to planes in $\mathbb{R}^3$
论文作者
论文摘要
令$γ：[0,1] \ rightArrow \ Mathbb {s}^{2} $是$ \ Mathbb {r}^3 $中的非分级曲线，也就是说，$ \ det \ det \ big big（γ（θ），γ'（θ），γ'（θ），γ），γ”（θ）\ big。令$v_θ=γ（θ）^\ perp $，让$π_θ：\ mathbb {r}^3 \ rightarrowv_θ$是正交投影$ \ text {dim}（π_θ（a））= \ min \ {2，\ text {dim} a \} $。 \ text {dim}（π_θ（a））<s \} $ $ \ MATHCAL {H}^2（π_θ（a））> 0 $。
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 $V_θ=γ(θ)^\perp$ and let $π_θ:\mathbb{R}^3\rightarrow V_θ$ be the orthogonal projections. We prove that if $A\subset \mathbb{R}^3$ is a Borel set, then for a.e. $θ\in [0,1]$ we have $\text{dim}(π_θ(A))=\min\{2,\text{dim} A\}$. More generally, we prove an exceptional set estimate. For $A\subset\mathbb{R}^3$ and $0\le s\le 2$, define $E_s(A):=\{θ\in[0,1]: \text{dim}(π_θ(A))<s\}$. 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. $θ\in[0,1]$ we have $\mathcal{H}^2(π_θ(A))>0$.
