# Sarnak's saturation problem for complete intersections

## TL;DR

This paper advances the understanding of almost prime solutions to Diophantine systems by significantly improving bounds on prime divisors, utilizing advanced sieve techniques and the circle method.

## Contribution

It generalizes the Br"udern-Fouvry vector sieve to higher dimensions and incorporates smooth weights into the circle method, improving bounds from polynomial to logarithmic growth.

## Key findings

- Bound on prime divisors improved to logarithmic scale
- Generalization of vector sieve to higher dimensions
- Enhanced circle method with smooth weights

## Abstract

We study almost prime solutions of systems of Diophantine equations in the Birch setting. Previous work shows that there exist integer solutions of size B with each component having no prime divisors below $B^{1/u}$, where $u=c_0n^{3/2}$, $n$ is the number of variables and $c_0$ is a constant depending on the degree and the number of equations. We improve the polynomial growth $$n^{3/2}$$ to the logarithmic $$\frac{\log n}{\log \log n}.$$ Our main new ingredients are the generalisation of the Br\"udern-Fouvry vector sieve in any dimension and the incorporation of smooth weights into the Davenport-Birch version of the circle method.