For almost all pairs of points in P^2(F_q), an element of SL3(Z) with entries bounded by q^{1/3+ε} realizes the projective action.
Best possible rates of distribution of dense lattice orbits in homo- geneous spaces
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Optimal Lifting for the Projective Action of $SL_3(Z)$
For almost all pairs of points in P^2(F_q), an element of SL3(Z) with entries bounded by q^{1/3+ε} realizes the projective action.