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

Abstract: Let

e p s i l o n > 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

g a m m a

in

S L_{3} (Z)

, with coordinates bounded by

q^{1 / 3 + e p s i l o n}

, whose projection to

S L_{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

S L_{2} (Z)

.

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