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