Apollonian circle packings are infinite configurations of circles generated starting from a given Descartes configuration of four mutually touching circles in the plane by inversions in circles though three tangency points of a configuration. A Descartes configuration a specified up to a Euclidean motion by the curvatures \((a,b,c,d)\) of the four circles. These satisfy the Descartes relation \(Q(a,b,c,d)=0\) where \(Q(a,b,c,d) = 2((a^2+b^2+c^2+d^2) - (a+b+c+d)^2\). The curvatures of Descartes configurations (\(a',b',c',d'\)) in a packing is described by the orbit of an action of a group of \(4 \times 4\) integer matrices, the Apollonian group \(A\) contained in \(\text{GL}(4,\mathbb Z)\). This fact is implicit in K. E. Hirst [J. Lond. Math. Soc. 42, 281–291 (1967; Zbl 0147.39102)] and explicit in B. Söderberg [Phys. Rev. A 46, No. 4, 1859–1866 (1992)] The group \(A\) is a ‘thin group’ in the sense that it is of infinite index in \(\text{GL}(4,\mathbb Z)\). This paper studies integer orbits of the group, and the congruence classes \((a', b', c', d')\) (modulo \(m\)) hit by such orbits, restricting to primitive orbits, ones where \(\gcd(a, b,c,d)=1\). Previously R. L. Graham et al. [J. Number Theory 100, No. 1, 1–45 (2003; Zbl 1026.11058)] showed that any primitive integer packing contains a curvature with residue classes (mod \(m\)). The main result of the present paper is to show that for a fixed primitive packing if \(P_m\) is the set of vectors of possible residues modulo \(m\) in that packing, writing \(m=d_1d_2\) with \((d_2,6)=1\) and \(d_1=2^f 3^g\), then:
(i) For \(d_1> 1\) the natural projection \(P_m \to P_{d_1} \times P_{d_2}\) is surjective;
(ii) The projection \(P_{d_2} \to \prod_{p^r || d_2} P_{p^r}\) is surjective and \(P_{p^r}=C_r\), where \(C_r\) is the set of all allowable values of all primitive integer points in the cone of integer points satisfying \(Q(a, b,c, d)=0\) (so is independent of the integer packing);
(iii) If \(f, g \geq 1\) then the natural projection \(P_{d_1} \to P_{2^f} \times P_{3^g}\) is surjective;
(iv) If \(f \geq 4\), then \(\pi: C_{2^f} \to C_{8}\) has \(P_{2^f} = \pi^{-1}{P_8}\);
(v) If \(g \geq 2\), let \(\phi: C_{3^g} \to C_{3}\) be the natural projection. Then \(P_{3^g} = \phi^{-1}(P_3)\).
This result shows that the complete congruence structure of an orbit is determined its values modulo \(24\). This result is important in applying the affine linear sieve of J. Bourgain, A. Gamburd and P. Sarnak [Invent. Math. 179, No. 3, 559–644 (2010; Zbl 1239.11103)] to analyze occurrence of integers with few prime divisors in Apollonian groups orbit. The proof of this result studies the preimage group \(\Gamma\) of \(A\) in the spin double cover of the automorphism group \(\text{SO}_{Q}\) of the Descartes quadratic form. It uses L. E. Dickson’s classification of subgroups of \(\text{SL}_2\) over finite fields, and Goursat’s lemma to handle squarefree \(m\). Elsewhere the author with K. Sanden [Exp. Math. 20, No. 4, 380–399 (2011; Zbl 1259.11065)] conjectured a local-to-global principle for lifting congruence information to every preimage integer orbit configuration of sufficiently large height in the group.