2018/12/18 by de Courcy-Ireland, Matthew, Lee, Seungjae · 1 citation
#05C50 #11D25 #11F72 #37P25 #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.1812.07275
We confirm, for the primes up to 3000, the conjecture of Bourgain, Gamburd, and Sarnak on strong approximation for the Markoff surface x2+y2+z2 = 3xyz modulo primes. For primes congruent to 3 modulo 4, we find data suggesting that some natural graphs constructed from this equation are asymptotically Ramanujan. For primes congruent to 1 modulo 4, the data suggest a weaker spectral gap. In both cases, there is close agreement with the Kesten-McKay law for the density of states for random 3-regular graphs. We also study the connectedness of other level sets x2+y2+z2-3xyz = k. In the degenerate case of the Cayley cubic, we give a complete description of the orbits.