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arXiv:1205.4416math.NT

On the Local-Global Conjecture for integral Apollonian gaskets

Jean Bourgain, Alex Kontorovich

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Abstract

We prove that a set of density one satisfies the local-global conjecture for integral Apollonian gaskets. That is, for a fixed integral, primitive Apollonian gasket, almost every (in the sense of density) admissible (passing local obstructions) integer is the curvature of some circle in the gasket.

arXiv comments
64 pages, 3 figures; Appendix by Peter Varju
Journal reference
DOI
Subjects
math.NTNumber Theory

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  1. Students’ insight proves that the local-global conjecture doesn’t hold 1 comment

    • That the curvatures are integers is proved (noticed by Soddy in 1936 as "a fairly straightforward consequence of Descartes’ equation", as the article mentions), and a lot more is also proved about the integers' distribution; the conjecture specifically is about whether every…

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