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$$x_1^2+x_2^2+\cdots+x_n^2=ax_1x_2\cdots x_n+k.$$When $n\ge 4$, 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 $n\ge 4$. 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.