轨道上的素数点：Bourgain-Gamburd-Sarnak猜想的一些证明
Prime Points in Orbits: Some Instances of the Bourgain-Gamburd-Sarnak Conjecture
摘要：
利用Vinogradov三素数定理的Vaughan变分证明了几类无限超曲面中的素点的Zurkki-稠密性，包括某些二次型的水平集、永久多项式和某些准齐次向量空间的定义多项式.其中三个家庭的情况下，一个猜想布尔甘，甘布尔和Sarnak关于总理点轨道的简单代数群。我们的方法是基于制定的一般条件上的定义多项式的超曲面，这足以保证Zurki密度的素数点是等价的存在一个奇点。
英文摘要：
We use Vaughan's variation on Vinogradov's three-primes theorem to prove Zariski-density of prime points in several infinite families of hypersurfaces, including level sets of some quadratic forms, the Permanent polynomial, and the defining polynomials of some pre-homogeneous vector spaces. Three of these families are instances of a conjecture by Bourgain, Gamburd and Sarnak regarding prime points in orbits of simple algebraic groups. Our approach is based on the formulation of a general condition on the defining polynomial of a hypersurface, which suffices to guarantee that Zariski-density of prime points is equivalent to the existence of an odd point.
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