An asymptotic formula for integer points on Markoff-Hurwitz varieties

Abstract

We establish an asymptotic formula for the number of integer solutions to the Markoff-Hurwitz equation
\[
x_{1}^{2}+x_{2}^{2}+\ldots+x_{n}^{2}=ax_{1}x_{2}\ldots x_{n}+k.
\]
When $n\geq4$ the previous best result is by Baragar (1998) that gives an exponential rate of growth with exponent $\beta$ that is not in general an integer when $n\geq4$. 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.

Authors

Alex Gamburd

CUNY Graduate Center, New York, NY, USA

Michael Magee

Durham University, Durham, UK

Ryan Ronan

Baruch College (CUNY), New York, NY, USA