Sarnak's saturation problem for complete intersections

We study almost prime solutions of systems of Diophantine equations in the Birch setting. Previous work shows that there exist integer solutions of size with each component having no prime divisors below , where equals , is the number of variables and is a constant depending on the degree and the number of equations. We improve the polynomial growth to the logarithmic . Our main new ingredients are the generalization of the Brüdern-Fouvry vector sieve in any dimension and the incorporation of smooth weights into the Davenport-Birch version of the circle method.