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

128 pages. Minor changes and corrections

References in corpus (1)

• Extending linear and quadratic functions from high rank varieties

Cited by in corpus (2)

• An inverse theorem for Freiman multi-homomorphisms
• Quantitative bounds for the $U^4$-inverse theorem over low characteristic finite fields