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

TL;DR

This paper proves that for large primes, with high probability, there exists a matrix in $SL_3(Z)$ with bounded size that maps one point to another in the projective plane over a finite field, extending strong approximation results.

Contribution

It establishes an optimal bound for lifting points in the projective plane over finite fields via $SL_3(Z)$, generalizing Sarnak's theorem from $SL_2(Z)$ to higher rank.

Findings

Existence of such matrices with size bounded by $q^{1/3+ ext{epsilon}}$

The exponent $1/3$ is proven to be optimal

High probability results for large primes

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 $SL_2(Z)$.