On restricted projections to planes 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 $V_θ=γ(θ)^\perp$ 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 $\mathcal{H}^2(π_θ(A))>0$.

Figures (2)

• Figure 1: High-low decomposition for $P_\theta$
• Figure 2: Relation between planks

Theorems & Definitions (46)

• Theorem 1
• Corollary 1
• Theorem 2
• Theorem 3: Marstrand's projection theorem
• Definition 1
• Proposition 1
• proof
• Definition 2: $(\delta,s)$-set
• Lemma 1
• Theorem 4