Affine linear sieve, expanders, and sum-product

Springer Science and Business Media LLC - Tập 179 - Trang 559-644 - 2009
Jean Bourgain1, Alex Gamburd2, Peter Sarnak1,3
1School of Mathematics, Institute for Advanced Study, Princeton, USA
2Department of Mathematics, University of California at Santa Cruz, Santa Cruz, USA
3Department of Mathematics, Princeton University, Princeton, USA

Tóm tắt

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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