AN INVERSE THEOREM FOR THE GOWERS Us+1[N]-NORM

作者：Green Ben^*; Tao Terence; Ziegler Tamar

来源：Electronic Research Announcements in Mathematical Sciences, 2011, 18: 69-90.

DOI：10.3934/era.2011.18.69

• Gowers norms
• nilsequences

This is an announcement of the proof of the inverse conjecture for the Gowers Us+1[ N]-norm for all s >= 3; this is new for s >= 4, the cases s = 1, 2, 3 having been previously established. More precisely we outline a proof 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) which 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. In particular, one obtains an asymptotic formula for the number of k-term arithmetic progressions p(1) < p(2) < . . . < p(k) <= N of primes, for every k >= 3.

• 出版日期2011