Prime Points in Orbits: Some Instances of the Bourgain-Gamburd-Sarnak Conjecture

2017/07/15 by Horesh, Tal, Nevo, Amos
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1707.04746

Abstract

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.

Related

• A note on Zariski dense orbit conjecture
• Prime and almost prime integral points on principal homogeneous spaces
• Effectivity for bounds on points of good reduction in the moduli space of hypersurfaces
• On the Density of Integer Points on the Generalised Markoff-Hurwitz and Dwork Hypersurfaces
• Prime divisors in polynomial orbits over function fields