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arXiv · 1705.09133

Sarnak's saturation problem for complete intersections

Damaris Schindler·Efthymios Sofos
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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.

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Damaris Schindler, Efthymios Sofos. 2017-05-25. Sarnak's saturation problem for complete intersections. https://doi.org/10.1112/s002557931800030x

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