Affine linear sieve, expanders, and sum-product

作者：Bourgain Jean; Gamburd Alex; Sarnak Peter^*

来源：Inventiones Mathematicae, 2010, 179(3): 559-644.

DOI：10.1007/s00222-009-0225-3

Let O be an orbit in Z(n) of a finitely generated subgroup Lambda of GL (n) (Z) whose Zariski closure Zcl(Lambda) 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(Lambda) = SL2, using crucially sum-product theorem in Z/qaZ sign for q square-free.

• 出版日期2010-3