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Bulletin of the American Mathematical Society
Bulletin of the American Mathematical Society
ISSN 1088-9485(online) ISSN 0273-0979(print)

 

From Apollonius to Zaremba: Local-global phenomena in thin orbits


Author: Alex Kontorovich
Journal: Bull. Amer. Math. Soc. 50 (2013), 187-228
MSC (2010): Primary 11F41, 11J70, 11P55, 20H10, 22E40
Published electronically: January 18, 2013
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Abstract: We discuss a number of natural problems in arithmetic, arising in completely unrelated settings, which turn out to have a common formulation involving ``thin'' orbits. These include the local-global problem for integral Apollonian gaskets and Zaremba's Conjecture on finite continued fractions with absolutely bounded partial quotients. Though these problems could have been posed by the ancient Greeks, recent progress comes from a pleasant synthesis of modern techniques from a variety of fields, including harmonic analysis, algebra, geometry, combinatorics, and dynamics. We describe the problems, partial progress, and some of the tools alluded to above.


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Additional Information

Alex Kontorovich
Affiliation: Department of Mathematics, Yale University, New Haven, Connecticut
Email: alex.kontorovich@yale.edu

DOI: http://dx.doi.org/10.1090/S0273-0979-2013-01402-2
PII: S 0273-0979(2013)01402-2
Received by editor(s): August 15, 2012
Received by editor(s) in revised form: November 4, 2012
Published electronically: January 18, 2013
Additional Notes: Partially supported by NSF grants DMS-1209373, DMS-1064214 and DMS-1001252.
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.