An illustrated introduction to the arithmetic of Apollonian circle packings, continued fractions, and other thin orbits

The paper proves strong approximation for the geometric Apollonian group by a Gaussian-integer (Z[i]) construction that produces all upper and lower triangular unipotents and then uses a counting argument to conclude surjectivity onto SL2(Z/p^mZ) for p ≥ 5. The candidate solution proves the same result by extracting explicit integral unipotents U(2) and L(4) from real generators of Ageo, reducing modulo p^m to get E12(1) and E21(1), and then giving a constructive elimination over the local ring Z/p^mZ. Both arguments are valid; the approaches are materially different but consistent. The paper’s statements about generators and the theorem itself align with the model’s steps and hypotheses.