Affine linear sieve, expanders, and sum-product

• 成果类型: Article
• 署名作者: Bourgain, Jean; Gamburd, Alex; Sarnak, Peter
• 署名单位: Princeton University; University of California System; University of California Santa Cruz; Institute for Advanced Study - USA
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-009-0225-3
• 发表日期: 2010
• 页码: 559-644
• 关键词: SUBGROUPS; number; FIELDS; set

摘要

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.