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

TL;DR

This paper proves the Zariski-density of prime points in various hypersurfaces, including those related to a conjecture by Bourgain, Gamburd, and Sarnak, using a novel approach based on prime number theorems.

Contribution

It introduces a general condition on hypersurface polynomials that links prime point density to the existence of an odd point, advancing understanding of prime orbits in algebraic groups.

Findings

Proves Zariski-density of prime points in several hypersurfaces.

Establishes equivalence between prime density and odd points under certain conditions.

Includes instances related to the Bourgain-Gamburd-Sarnak conjecture.

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.