On the Local-Global Principle for Integral Apollonian-3 Circle Packings

TL;DR

This paper investigates the integral properties of Apollonian-3 circle packings, proposing a local-global conjecture, and proves a density one result along with a spectral gap for related symmetry groups.

Contribution

It introduces a local-global conjecture for Apollonian-3 packings, proves a density one version, and establishes a uniform spectral gap for congruence towers.

Findings

Proved a density one version of the local-global conjecture.

Established a uniform spectral gap for congruence towers.

Analyzed the reduction theory of Apollonian-3 packings.

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 congruence towers of the symmetry group.