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

Abstract

Let gamma :[0,1]rightarrow mathbb S^{2} be a non-degenerate curve in mathbb R^3, that is to say, det big (gamma (theta ),gamma '(theta ),gamma ''(theta )big )ne 0. For each theta in [0,1], let l_theta =text {span}(gamma (theta )) and rho _theta :mathbb R^3rightarrow l_theta be the orthogonal projections. We prove an exceptional set estimate. For any Borel set Asubset mathbb R^3 and 0le sle 1, define E_s(A):={theta in [0,1]: dim (rho _theta (A))<s}. We have dim (E_s(A))le max {0,1+frac{s-dim (A)}{2}}.