Search for: All records

In a primitive integral Apollonian circle packing, the curvatures that appear must fall into one of six or eight residue classes modulo 24. The local-global conjecture states that every sufficiently large integer in one of these residue classes appears as a curvature in the packing. We prove that this conjecture is false for many packings, by proving that certain quadratic and quartic families are missed. The new obstructions are a property of the thin Apollonian group (and not its Zariski closure), and are a result of quadratic and quartic reciprocity, reminiscent of a Brauer-Manin obstruction. Based on computational evidence, we formulate a new conjecture. 

Abstract A circle of curvature $$n\in \mathbb{Z}^+$$ is a part of finitely many primitive integral Apollonian circle packings. Each such packing has a circle of minimal curvature $$-c\leq 0$$, and we study the distribution of $c/n$ across all primitive integral packings containing a circle of curvature $$n$$. As $$n\rightarrow \infty $$, the distribution is shown to tend towards a picture we name the Apollonian staircase. A consequence of the staircase is that if we choose a random circle packing containing a circle $$C$$ of curvature $$n$$, then the probability that $$C$$ is tangent to the outermost circle tends towards $$3/\pi $$. These results are found by using positive semidefinite quadratic forms to make $$\mathbb{P}^1(\mathbb{C})$$ a parameter space for (not necessarily integral) circle packings. Finally, we examine an aspect of the integral theory known as spikes. When $$n$$ is prime, the distribution of $c/n$ is extremely smooth, whereas when $$n$$ is composite, there are certain spikes that correspond to prime divisors of $$n$$ that are at most $$\sqrt{n}$$. 

A practical algorithm to compute the fundamental domain of an arithmetic Fuchsian group was given by Voight, and implemented in Magma. It was later expanded by Page to the case of arithmetic Kleinian groups. We combine and improve on parts of both algorithms to produce a more efficient algorithm for arithmetic Fuchsian groups. This algorithm is implemented in PARI/GP, and we demonstrate the improvements by comparing running times versus the live Magma implementation.