An asymptotic formula for integer points on Markoff-Hurwitz varieties

• 成果类型: Article
• 署名作者: Gamburd, Alex; Magee, Michael; Ronan, Ryan
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2019.190.3.2
• 发表日期: 2019
• 页码: 751-809
• 关键词: hausdorff dimension; simple geodesics; number; DYNAMICS; GROWTH; analog

摘要

We establish an asymptotic formula for the number of integer solutions to the Markoff-Hurwitz equation x(1)(2) + x(2)(2) + ... +x(n)(2) = ax(1)x(2) ... x(n) + k. When n >= 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 >= 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.