Sharp estimates for Gowers norms on discrete cubes

TL;DR

This paper derives sharp estimates for Gowers norms on discrete cubes, establishing explicit formulas and asymptotic behaviors for key exponents related to these inequalities, extending previous results in the field.

Contribution

It provides a unified theory linking critical exponents in Gowers norm inequalities, generalizes known theorems, and offers precise asymptotic formulas for these exponents.

Findings

Explicit formula for $t_{k,2}$ generalizing Kane and Tao's theorem.

Two-sided asymptotic estimates for $t_{k,n}$ as $n oigcirc$ for fixed $k$.

Precise asymptotic formula for $t_{k,n}$ as $k oigcirc$ for fixed $n$.

Abstract

We study optimal dimensionless inequalities $$ \|f\|_{U^k} \leq \|f\|_{\ell^{p_{k,n}}} $$ that hold for all functions $f\colon\mathbb{Z}^d\to\mathbb{C}$ supported in $\{0,1,\ldots,n-1\}^d$ and estimates $$ \|1_A\|_{U^k}^{2^k}\leq |A|^{t_{k,n}} $$ that hold for all subsets $A$ of the same discrete cubes. A general theory, analogous to the work of de Dios Pont, Greenfeld, Ivanisvili, and Madrid, is developed to show that the critical exponents are related by $p_{k,n} t_{k,n} = 2^k$. This is used to prove the three main results of the paper: an explicit formula for $t_{k,2}$, which generalizes a theorem by Kane and Tao, two-sided asymptotic estimates for $t_{k,n}$ as $n\to\infty$ for a fixed $k\geq2$, which generalize a theorem by Shao, and a precise asymptotic formula for $t_{k,n}$ as $k\to\infty$ for a fixed $n\geq2$.