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

TL;DR

This paper presents a new proof for the inverse theorem of Gowers $U^{s+1}$-norms over cyclic groups, providing explicit quantitative bounds without relying on regularity or non-standard analysis, especially for $s \,\geq\, 3$.

Contribution

It introduces a novel proof technique that yields explicit bounds and avoids traditional complex methods for the inverse theorem in this setting.

Findings

Provides a new proof with double-exponential bounds

Eliminates the need for regularity or non-standard analysis

Applicable for $s \ge 3$ in cyclic groups

Abstract

We provide a new proof of the inverse theorem for the Gowers $U^{s+1}$-norm over groups $H=\mathbb Z/N\mathbb Z$ 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.