On the local-global principle for integral Apollonian 3-circle packings

Abstract

Abstract In this paper we study the integral properties of Apollonian 3-circle packings, which are variants of the standard Apollonian circle packings. Specifically, we study the reduction theory, formulate a local-global conjecture, and prove a density one version of this conjecture. Along the way, we prove a uniform spectral gap for the congruence towers of the symmetry group.