An asymptotic formula for integer points on Markoff-Hurwitz varieties

TL;DR

This paper derives an asymptotic formula for counting integer solutions to the Markoff-Hurwitz equation, extending previous results by providing a new interpretation of the growth exponent through conformal measures.

Contribution

It introduces a new asymptotic counting formula for solutions to the Markoff-Hurwitz equation and offers a novel interpretation of the growth exponent via conformal measures.

Findings

Established an asymptotic formula for integer solutions

Connected growth exponent to conformal measure parameter

Extended previous exponential growth results for n ≥ 4

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 $n\geq 4$. 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.