Affine linear sieve, expanders, and sum-product
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## Affine linear sieve, expanders, and sum-product

• Author(s): Bourgain, Jean
• Gamburd, Alex
• Sarnak, Peter
• et al.

## Published Web Location

https://doi.org/10.1007/s00222-009-0225-3
Abstract

Let $\mathcal{O}$ be an orbit in ℤ n of a finitely generated subgroup Λ of GL n (ℤ) whose Zariski closure Zcl(Λ) is suitably large (e.g. isomorphic to SL2). We develop a Brun combinatorial sieve for estimating the number of points on $\mathcal{O}$ at which a fixed integral polynomial is prime or has few prime factors, and discuss applications to classical problems, including Pythagorean triangles and integral Apollonian packings. A fundamental role is played by the expansion property of the “congruence graphs” that we associate with $\mathcal{O}$ . This expansion property is established when Zcl(Λ)=SL2, using crucially sum-product theorem in ℤ/qℤ for q square-free.

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