## A proof of the positive density conjecture for integer Apollonian circle packings

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- by Jean Bourgain and Elena Fuchs;
- J. Amer. Math. Soc.
**24**(2011), 945-967 - DOI: https://doi.org/10.1090/S0894-0347-2011-00707-8
- Published electronically: June 20, 2011
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## Abstract:

An Apollonian circle packing (ACP) is an ancient Greek construction which is made by repeatedly inscribing circles into the triangular interstices in a Descartes configuration of four mutually tangent circles. Remarkably, if the original four circles have integer curvature, all of the circles in the packing will have integer curvature as well. In this paper, we compute a lower bound for the number $\kappa (P,X)$ of integers less than $X$ occurring as curvatures in a bounded integer ACP $P$, and prove a conjecture of Graham, Lagarias, Mallows, Wilkes, and Yan that the ratio $\kappa (P,X)/X$ is greater than $0$ for $X$ tending to infinity.## References

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## Bibliographic Information

**Jean Bourgain**- Affiliation: Institute for Advanced Study, School of Mathematics, Einstein Drive, Princeton, New Jersey 08540
- MR Author ID: 40280
- Email: bourgain@math.ias.edu
**Elena Fuchs**- Affiliation: Institute for Advanced Study, School of Mathematics, Einstein Drive, Princeton, New Jersey 08540
- Email: efuchs@math.ias.edu
- Received by editor(s): January 21, 2010
- Received by editor(s) in revised form: February 24, 2011, and June 6, 2011
- Published electronically: June 20, 2011
- Additional Notes: The first author is supported in part by NSF grant DMS–0808042

The second author was supported in part by NSF grant DMS–0635607 - © Copyright 2011 American Mathematical Society
- Journal: J. Amer. Math. Soc.
**24**(2011), 945-967 - MSC (2010): Primary 11D09, 11E16, 11E20
- DOI: https://doi.org/10.1090/S0894-0347-2011-00707-8
- MathSciNet review: 2813334