An asymptotic formula for integer points on Markoff-Hurwitz varieties

2016/03/20 by Gamburd, Alex, Magee, Michael, Ronan, Ryan
#11D25 #11D41 #11D45 #11J70 #37D25 #37D35 #Dynamical Systems (math.DS) #FOS: Mathematics #Group Theory (math.GR) #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1603.06267

Abstract

We establish an asymptotic formula for the number of integer solutions to the Markoff-Hurwitz equation x₁²+x₂²+…+x_n²=ax₁x₂… x_n+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.

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