Geodesic
Quantitative bounds for the $U^4$-inverse theorem over low characteristic finite fields
Discrete analysis (2022)

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arXiv
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 $p\geq 5$. 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 $k,p$.
Publié le : 
Jonathan Tidor. Quantitative bounds for the $U^4$-inverse theorem over low characteristic finite fields. Discrete analysis (2022). http://geodesic.mathdoc.fr/item/DAS_2022_a6/
@article{DAS_2022_a6,
     author = {Jonathan Tidor},
     title = {Quantitative bounds for the $U^4$-inverse theorem over low characteristic finite fields},
     journal = {Discrete analysis},
     year = {2022},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DAS_2022_a6/}
}
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