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SARNAK’S SATURATION PROBLEM FOR COMPLETE INTERSECTIONS

Part of: Additive number theory; partitions Diophantine equations

Published online by Cambridge University Press:  24 August 2018

D. Schindler  and
E. Sofos
D. Schindler
Affiliation:
Universiteit Utrecht, Mathematisch Instituut, Budapestlaan 6, Utrecht, 3584 CD, Netherlands email [email protected]
E. Sofos
Affiliation:
Max Planck Institute for Mathematics, Vivatsgasse 7, Bonn, 53111, Germany email [email protected]
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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 $u$ equals $c_{0}n^{3/2}$, $n$ 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 $(\log n)(\log \log n)^{-1}$. 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.

MSC classification

Primary: 11D72: Equations in many variables
Secondary: 11P32: Goldbach-type theorems; other additive questions involving primes
Type
Research Article
Information
Mathematika , Volume 65 , Issue 1 , 2019 , pp. 1 - 56
Copyright
Copyright © University College London 2018 

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