Geodesic
Quantitative inverse theorem for Gowers uniformity norms $\mathsf {U}^5$ and $\mathsf {U}^6$ in $\mathbb {F}_2^n$
Canadian journal of mathematics, Tome 76 (2024) no. 4, pp. 1289-1338

Voir la notice de l'article provenant de la source Cambridge

DOI

We prove quantitative bounds for the inverse theorem for Gowers uniformity norms $\mathsf {U}^5$ and $\mathsf {U}^6$ in $\mathbb {F}_2^n$. The proof starts from an earlier partial result of Gowers and the author which reduces the inverse problem to a study of algebraic properties of certain multilinear forms. The bulk of the work in this paper is a study of the relationship between the natural actions of $\operatorname {Sym}_4$ and $\operatorname {Sym}_5$ on the space of multilinear forms and the partition rank, using an algebraic version of regularity method. Along the way, we give a positive answer to a conjecture of Tidor about approximately symmetric multilinear forms in five variables, which is known to be false in the case of four variables. Finally, we discuss the possible generalization of the argument for $\mathsf {U}^k$ norms.
DOI : 10.4153/S0008414X23000391
Mots-clés : Gowers uniformity norms, multilinear forms, symmetric groups, partition rank, inverse theorems 
Milićević, Luka. Quantitative inverse theorem for Gowers uniformity norms $\mathsf {U}^5$ and $\mathsf {U}^6$ in $\mathbb {F}_2^n$. Canadian journal of mathematics, Tome 76 (2024) no. 4, pp. 1289-1338. doi: 10.4153/S0008414X23000391
@article{10_4153_S0008414X23000391,
     author = {Mili\'cevi\'c, Luka},
     title = {Quantitative inverse theorem for {Gowers} uniformity norms $\mathsf {U}^5$ and $\mathsf {U}^6$ in $\mathbb {F}_2^n$},
     journal = {Canadian journal of mathematics},
     pages = {1289--1338},
     year = {2024},
     volume = {76},
     number = {4},
     doi = {10.4153/S0008414X23000391},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/S0008414X23000391/}
}
TY  - JOUR
AU  - Milićević, Luka
TI  - Quantitative inverse theorem for Gowers uniformity norms $\mathsf {U}^5$ and $\mathsf {U}^6$ in $\mathbb {F}_2^n$
JO  - Canadian journal of mathematics
PY  - 2024
SP  - 1289
EP  - 1338
VL  - 76
IS  - 4
UR  - http://geodesic.mathdoc.fr/articles/10.4153/S0008414X23000391/
DO  - 10.4153/S0008414X23000391
ID  - 10_4153_S0008414X23000391
ER  - 
%0 Journal Article
%A Milićević, Luka
%T Quantitative inverse theorem for Gowers uniformity norms $\mathsf {U}^5$ and $\mathsf {U}^6$ in $\mathbb {F}_2^n$
%J Canadian journal of mathematics
%D 2024
%P 1289-1338
%V 76
%N 4
%U http://geodesic.mathdoc.fr/articles/10.4153/S0008414X23000391/
%R 10.4153/S0008414X23000391
%F 10_4153_S0008414X23000391

Cité par Sources :