Optimal Lifting for the Projective Action of $SL_3(Z)$

Abstract

Let $\epsilon>0$ and let $q$ be a prime going to infinity. We prove that with high probability given $x,y$ in the projective plane over the finite field $F_q$ there exists $\gamma$ in $SL_3(Z)$, with coordinates bounded by $q^{1/3+\epsilon}$, whose projection to $SL_{3}(F_q)$ sends $x$ to $y$. The exponent $1/3$ is optimal and the result is a high rank generalization of Sarnak's optimal strong approximation theorem for $SL_2(Z)$.