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arXiv · 1707.04746

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

Tal Horesh·Amos Nevo
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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.

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Tal Horesh, Amos Nevo. 2017-07-15. Prime Points in Orbits: Some Instances of the Bourgain-Gamburd-Sarnak Conjecture. https://arxiv.org/abs/1707.04746

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