Quasipolynomial bounds on the inverse theorem for the Gowers U^s+1[N]-norm

We prove quasipolynomial bounds on the inverse theorem for the Gowers U^s+1[N]-norm. The proof is modeled after work of Green, Tao, and Ziegler and uses as a crucial input recent work of the first author regarding the equidistribution of nilsequences. In a companion paper, this result will be used to improve the bounds on Szemerédi's theorem.