An Exceptional Set Estimate for Restricted Projections to Lines in $$\mathbb R^3$$

Abstract

Let $$\gamma :[0,1]\rightarrow \mathbb S^{2}$$ γ : [ 0 , 1 ] → S 2 be a non-degenerate curve in $$\mathbb R^3$$ R 3 , that is to say, $$\det \big (\gamma (\theta ),\gamma '(\theta ),\gamma ''(\theta )\big )\ne 0$$ det ( γ ( θ ) , γ ′ ( θ ) , γ ′ ′ ( θ ) ) ≠ 0 . For each $$\theta \in [0,1]$$ θ ∈ [ 0 , 1 ] , let $$l_\theta =\text {span}(\gamma (\theta ))$$ l θ = span ( γ ( θ ) ) and $$\rho _\theta :\mathbb R^3\rightarrow l_\theta $$ ρ θ : R 3 → l θ be the orthogonal projections. We prove an exceptional set estimate. For any Borel set $$A\subset \mathbb R^3$$ A ⊂ R 3 and $$0\le s\le 1$$ 0 ≤ s ≤ 1 , define $$E_s(A):=\{\theta \in [0,1]: \dim (\rho _\theta (A))<s\}$$ E s ( A ) : = { θ ∈ [ 0 , 1 ] : dim ( ρ θ ( A ) ) < s } . We have $$\dim (E_s(A))\le \max \{0,1+\frac{s-\dim (A)}{2}\}$$ dim ( E s ( A ) ) ≤ max { 0 , 1 + s - dim ( A ) 2 } .

A note on Kakeya sets of horizontal and SL(2)$SL(2)$ lines

“…In fact, up to the rotation by $$45^{\circ }$$, this family of projections is precisely the ‘model example’ mentioned just below [3, (1)]. Therefore, (3.5) follows from [3, Corollary 1], and the proof is complete.$$\Box$$…”

“…y \in \mathbb {R}. \end{equation}$$The idea is that $$\lbrace \pi _{V_{y}}\rbrace _{y \in \mathbb {R}}$$ is a one‐parameter family of orthogonal projections to planes in $$\mathbb {R}^{3}$$ which satisfies the hypotheses of [3, Corollary 1]. Which planes are the planes $$V_{y}$$?…”

“…Moreover, the lines $$\ell _{y} := \operatorname{span}(e_{y})$$ are all contained in a $$45^{\circ }$$ rotated copy of the light cone $$$$\begin{equation*} \mathcal {C} := \lbrace (x,y,t) \in \mathbb {R}^{3} : t^{2} = x^{2} + y^{2}\rbrace , \end{equation*}$$$$see [2, Section 2.2] for the details. This implies that the projections $$\lbrace \pi _{V_{y}}\rbrace _{y \in \mathbb {R}}$$ satisfy the curvature condition [3, (1)]. In fact, up to the rotation by $$45^{\circ }$$, this family of projections is precisely the ‘model example’ mentioned just below [3, (1)].…”

“…This implies that the projections $$\lbrace \pi _{V_{y}}\rbrace _{y \in \mathbb {R}}$$ satisfy the curvature condition [3, (1)]. In fact, up to the rotation by $$45^{\circ }$$, this family of projections is precisely the ‘model example’ mentioned just below [3, (1)]. Therefore, (3.5) follows from [3, Corollary 1], and the proof is complete.$$\Box$$…”

“…These projection problems concern ‘restricted’ families of projections to planes in $$\mathbb {R}^{3}$$. Sharp results for such families were recently established by Gan, Guo, Guth, Harris, Maldague and Wang [3]. This is the second key component in the proof of Theorem 1.5.…”