Quantitative bounds in the inverse theorem for the Gowers $U^{s+1}$-norms over cyclic groups

We provide a new proof of the inverse theorem for the Gowers $U^{s+1}$-norm over groups $H=\mathbb{Z}\mathrm{/}N\mathbb{Z}$ for $N$ prime. This proof gives reasonable quantitative bounds (the worst parameters are double-exponential), and in particular does not make use of regularity or non-standard analysis, both of which are new for $s\ge 3$ in this setting.