From Apollonius to Zaremba: Local-global phenomena in thin orbits
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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 âthinâ orbits. These include the local-global problem for integral Apollonian gaskets and Zarembaâs 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.References
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Additional Information
- Alex Kontorovich
- Affiliation: Department of Mathematics, Yale University, New Haven, Connecticut
- MR Author ID: 704943
- ORCID: 0000-0001-7626-8319
- Email: alex.kontorovich@yale.edu
- 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.
- © Copyright 2013
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Bull. Amer. Math. Soc. 50 (2013), 187-228
- MSC (2010): Primary 11F41, 11J70, 11P55, 20H10, 22E40
- DOI: https://doi.org/10.1090/S0273-0979-2013-01402-2
- MathSciNet review: 3020826