Quantitative bounds for the $U^4$-inverse theorem over low characteristic finite fields

Abstract

This paper gives the first quantitative bounds for the inverse theorem for the Gowers $U^4$-norm over ${\mathbb{F}}_p^n$ when $p=2,3$. We build upon earlier work of Gowers and Milićević who solved the corresponding problem for $p\ge 5$. Our proof has two main steps: symmetrization and integration of low-characteristic trilinear forms. We are able to solve the integration problem for all $k$-linear forms, but the symmetrization problem we are only able to solve for trilinear forms. We pose several open problems about symmetrization of low-characteristic $k$-linear forms whose resolution, combined with recent work of Gowers and Milićević, would give quantitative bounds for the inverse theorem for the Gowers $U^{k+1}$-norm over ${\mathbb{F}}_p^n$ for all $k,p$.

arXiv comment17 pages

References in corpus (6)

• Polynomial bound for the partition rank vs the analytic rank of tensors
• On the Schmidt and analytic ranks for trilinear forms
• An inverse theorem for Freiman multi-homomorphisms
• A quantitative inverse theorem for the $U^4$ norm over finite fields
• Quantitative bounds in the inverse theorem for the Gowers $U^{s+1}$-norms over cyclic groups
• Non-classical polynomials and the inverse theorem