  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 Szemer\'{e}di's theorem.
