An inverse theorem for the Gowers Us+1[N]-norm

• 成果类型: Article
• 署名作者: Green, Ben; Tao, Terence; Ziegler, Tamar
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2012.176.2.11
• 发表日期: 2012
• 页码: 1231-1372
• 关键词: polynomial-sequences; uniform-distribution; multiple recurrence; ergodic averages; CONVERGENCE; EQUATIONS; szemeredi; BEHAVIOR; VALUES; PROOF

摘要

We prove the inverse conjecture for the Gowers Us+1[N]-norm for all s >= 1; this is new for s >= 4. More precisely, we establish that if f : [N] -> [-1, 1] is a function with parallel to f parallel to(Us+1) ([N]) >= delta, then there is a bounded-complexity s-step nilsequence F(g(n)Gamma) that correlates with f, where the bounds on the complexity and correlation depend only on s and delta. From previous results, this conjecture implies the Hardy-Littlewood prime tuples conjecture for any linear system of finite complexity.