Quasipolynomial inverse theorem for the U⁴(\mathbbF_pⁿ) norm

2024/10/11 by Luka Milićević, Milićević, Luka
Computer Science · Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #Matrix Theory and Algorithms #Number Theory (math.NT) #Numerical methods in inverse problems #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2410.08966

openalex publication_date 2024/10/11 · openalex created_date 2024/10/16 · openalex updated_date 2026/07/28

Abstract

The inverse theory for Gowers uniformity norms is one of the central topics in additive combinatorics and one of the most important aspects of the theory is the question of bounds. In this paper, we prove a quasipolynomial inverse theorem for the U⁴ norm in finite vector spaces. The proof follows a different strategy compared to the existing quantitative inverse theorems. In particular, the argument relies on a novel argument, which we call the abstract Balog-Szemerédi-Gowers theorem, and combines several other ingredients such as algebraic regularity method, bilinear Bogolyubov argument and algebraic dependent random choice.

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