Resumen de An inverse theorem for the Gowers U s+1 [N] -norm

Benjamin Green, Terence Tao, Tamar D. Ziegler

• We prove the inverse conjecture for the Gowers U s+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 ?f? U s+1 [N] =d , then there is a bounded-complexity s -step nilsequence F(g(n)G) that correlates with f , where the bounds on the complexity and correlation depend only on s and d . From previous results, this conjecture implies the Hardy-Littlewood prime tuples conjecture for any linear system of finite complexity.