An asymptotic formula for integer points on Markoff-Hurwitz varieties

Gamburd, Alex; Magee, Michael; Ronan, Ryan

doi:10.4007/annals.2019.190.3.2

An asymptotic formula for integer points on Markoff-Hurwitz varieties

Gamburd, Alex; Magee, Michael; Ronan, Ryan

Authors

Alex Gamburd

Professor Michael Magee michael.r.magee@durham.ac.uk
Professor

Ryan Ronan

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