Affine Sieve

TL;DR

This paper proves the main saturation conjecture related to applying a Brun sieve in affine linear group orbits, assuming the Zariski closure is Levi-semisimple, advancing understanding of sieve methods in algebraic group actions.

Contribution

It establishes the saturation conjecture for affine linear group orbits with Levi-semisimple Zariski closure, a significant step in sieve theory and algebraic groups.

Findings

Proves the main saturation conjecture in affine sieve setting.

Shows the Levi-semisimple condition is likely necessary.

Advances the application of Brun sieve in algebraic group orbits.

Abstract

We establish the main saturation conjecture in [BGS10] connected with executing a Brun sieve in the setting of an orbit of a group of affine linear transformations. This is carried out under the condition that the Zariski closure of the group is Levi-semisimple. It is likely that this condition is also necessary for such saturation to hold.