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

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