### Abstract

Let 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 SL_{2}). We develop a Brun combinatorial sieve for estimating the number of points on 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 O. This expansion property is established when Zcl(Λ)=SL_{2}, using crucially sum-product theorem in ℤ/qℤ for q square-free.

Original language | English (US) |
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Pages (from-to) | 559-644 |

Number of pages | 86 |

Journal | Inventiones Mathematicae |

Volume | 179 |

Issue number | 3 |

DOIs | |

State | Published - Jan 2010 |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)

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## Cite this

Bourgain, J., Gamburd, A., & Sarnak, P. (2010). Affine linear sieve, expanders, and sum-product.

*Inventiones Mathematicae*,*179*(3), 559-644. https://doi.org/10.1007/s00222-009-0225-3