Quantitative inverse bounds for the U^3 norm

Derive effective quantitative inverse-theorem bounds for the Gowers U^3 norm that can support efficient structure-testing algorithms, resolving the stated major open challenge in higher-order Fourier analysis.

Background

The paper explains that the complexity of its higher-degree quantum tests depends on quantitative inverse theorems converting correlation thresholds into explicit Gowers-norm bounds. Although quasipolynomial quantitative results are cited for several higher norms and settings, the authors identify the U3 case as a major unresolved quantitative problem.

References

However, for the U3 norm, obtaining quantitative bounds remains a major open challenge.

— Quantum Algorithms for Gowers Norm Estimation, Polynomial Testing, and Arithmetic Progression Counting over Finite Abelian Groups  (2508.01231 - Kuo, 2 Aug 2025) in Section 6, paragraph immediately preceding subsection “Counting for three terms Arithmetic Progressions in 𝔽_p^n”