Original inverse Gowers conjecture

Establish the original inverse Gowers conjecture: for every fixed prime p, integer d≥1, and ε>0, prove that any function F:𝔽_p^n→ℂ with ||F||∞≤1 and ||F||_{U^{d+1}}≥ε correlates by at least δ(p,d,ε)>0 with a degree-d polynomial phase e^{2πiP/p}.

Background

The paper introduces the original inverse Gowers conjecture as the proposed connection between a large Gowers U{d+1}-norm and correlation with a classical degree-d polynomial phase. The conjecture is known for d=1 and d=2, but the paper explains that generalized polynomials create complications for d=3 and higher and that the original formulation fails in that regime. It is nevertheless explicitly presented as the foundational conjectural statement motivating the discussion of inverse theorems.

References

We begin by discussing the Original Inverse Gowers Conjecture, which is central to understanding the connection between Gowers norms and polynomial structure:

— Quantum Algorithms for Gowers Norm Estimation, Polynomial Testing, and Arithmetic Progression Counting over Finite Abelian Groups  (2508.01231 - Kuo, 2 Aug 2025) in Section 2, subsection “Inverse Theorem”