Geodesic
An asymptotic formula for integer points on Markoff-Hurwitz varieties
Alex Gamburd1 ; Michael Magee2 ; Ryan Ronan3
1 CUNY Graduate Center, New York, NY, USA
2 Durham University, Durham, UK
3 Baruch College (CUNY), New York, NY, USA
Annals of mathematics, Tome 190 (2019) no. 3, pp. 751-809

Voir la notice de l'article provenant de la source Annals of Mathematics website

We establish an asymptotic formula for the number of integer solutions to the Markoff-Hurwitz equation \[ x_{1}^{2}+x_{2}^{2}+\cdots +x_{n}^{2}=ax_{1}x_{2}\cdots x_{n}+k. \] When $n\geq 4$, 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 $n\geq 4$. 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.

DOI : 10.4007/annals.2019.190.3.2

Alex Gamburd 1 ; Michael Magee 2 ; Ryan Ronan 3

1 CUNY Graduate Center, New York, NY, USA
2 Durham University, Durham, UK
3 Baruch College (CUNY), New York, NY, USA 
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Alex Gamburd; Michael Magee; Ryan Ronan. An asymptotic formula for integer points on Markoff-Hurwitz varieties. Annals of mathematics, Tome 190 (2019) no. 3, pp. 751-809. doi: 10.4007/annals.2019.190.3.2

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