Abstract

We confirm, for the primes up to 3000, the conjecture of Bourgain-Gamburd-Sarnak and Baragar on strong approximation for the Markoff surface $x^2+y^2+z^2=3xyz$ 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 $x^2+y^2+z^2-3xyz=k$. In the degenerate case of the Cayley cubic, we give a complete description of the orbits.

摘要

我们证实，对于高达 3000 的素数，Bourgain-Gamburd-Sarnak 和 Baragar 的猜想对马尔科夫曲面的强近似$X^2+{\mathrm{是的}}^2+z^2=3X\mathrm{是的}z$模素数。对于与 3 模 4 相等的素数，我们发现数据表明由该方程构造的一些自然图是渐近拉马努金的。对于与 1 模 4 一致的素数，数据表明光谱间隙较弱。在这两种情况下，随机 3-正则图的状态密度与 Kesten-McKay 定律非常一致。我们还研究了其他水平集的连通性$X^2+{\mathrm{是的}}^2+z^2-3X\mathrm{是的}z=ķ$. 在凯莱立方的简并情况下，我们给出了轨道的完整描述。