An Exceptional Set Estimate for Restricted Projections to Lines in ℝ3

Abstract

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

}

.