Alex Gamburd
An asymptotic formula for integer points on Markoff-Hurwitz varieties
Gamburd, Alex; Magee, Michael; Ronan, Ryan
Abstract
We establish an asymptotic formula for the number of integer solutions to the Markoff-Hurwitz equation x21+x22+⋯+x2n=ax1x2⋯xn+k. When n≥4, the previous best result is by Baragar (1998) that gives an exponential rate of growth with exponent β that is not in general an integer when n≥4. We give a new interpretation of this exponent of growth in terms of the unique parameter for which there exists a certain conformal measure on projective space.
Citation
Gamburd, A., Magee, M., & Ronan, R. (2019). An asymptotic formula for integer points on Markoff-Hurwitz varieties. Annals of Mathematics, 190(3), 751-809. https://doi.org/10.4007/annals.2019.190.3.2
Journal Article Type | Article |
---|---|
Acceptance Date | Jul 16, 2019 |
Online Publication Date | Oct 28, 2019 |
Publication Date | Nov 1, 2019 |
Deposit Date | Sep 22, 2017 |
Publicly Available Date | Aug 1, 2019 |
Journal | Annals of Mathematics |
Print ISSN | 0003-486X |
Electronic ISSN | 1939-8980 |
Publisher | Department of Mathematics |
Peer Reviewed | Peer Reviewed |
Volume | 190 |
Issue | 3 |
Pages | 751-809 |
DOI | https://doi.org/10.4007/annals.2019.190.3.2 |
Public URL | https://durham-repository.worktribe.com/output/1348708 |
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