An asymptotic formula for integer points on Markoff-Hurwitz varieties


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.


Alex Gamburd

CUNY Graduate Center, New York, NY, USA

Michael Magee

Durham University, Durham, UK

Ryan Ronan

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