Sarnak's saturation problem for complete intersections

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.