2023/11/20 by Elisa Bellah, Bellah, Elisa, Siran Chen +5
Mathematics · #Analytic Number Theory Research #FOS: Mathematics #Geometric and Algebraic Topology #Limits and Structures in Graph Theory #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2311.11468
openalex publication_date 2023/11/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In 2016, Bourgain, Gamburd, and Sarnak proved that Strong Approximation holds for the Markoff surface in most cases. That is, the modulo p solutions to the equation X12+X22+X32=3X1X2X3 are covered by the integer solutions for most primes p. In this paper, we provide upper bounds on lifts of mod p points of the Markoff surface by analyzing the growth along paths in the Markoff mod p graphs. Our first upper bound follows the algorithm given in the paper of Bourgain, Gamburd, and Sarnak, which constructs a path of possibly long length but where points grow relatively slowly. Our second bound considers paths in these graphs of short length but possibly large growth. We then provide numerical evidence and heuristic arguments for how these bounds might be improved.