Strong approximation in the Apollonian group

The Apollonian group is a finitely generated, infinite index subgroup of the orthogonal group OQ(Z) fixing the Descartes quadratic form Q. For nonzero v∈Z4 satisfying Q(v)=0, the orbits Pv=Av correspond to Apollonian circle packings in which every circle has integer curvature. In this paper, we specify the reduction of primitive orbits Pv mod any integer d>1. We show that this reduction has a multiplicative structure, and that mod primes p⩾5 it is the full cone of integer solutions to Q(v)≡0 for v≢0. This analysis is an essential ingredient in applications of the affine linear sieve as developed by Bourgain, Gamburd and Sarnak.