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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.
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The Apollonian Staircase
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 $$\sqrt{n}$$.
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- Award ID(s):
- 1652238
- PAR ID:
- 10447194
- Date Published:
- Journal Name:
- International Mathematics Research Notices
- ISSN:
- 1073-7928
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1093/imrn/rnad007
