Affine linear sieve, expanders, and sum-product

Jean Bourgain, Alex Gamburd, Peter Sarnak

Research output: Contribution to journalArticle

69 Scopus citations

Abstract

Let O be an orbit in ℤn of a finitely generated subgroup Λ of GLn(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 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)=SL2, using crucially sum-product theorem in ℤ/qℤ for q square-free.

Original languageEnglish (US)
Pages (from-to)559-644
Number of pages86
JournalInventiones Mathematicae
Volume179
Issue number3
DOIs
StatePublished - Jan 2010

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

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