An inverse theorem for Gowers norms of trace functions over prime fields

TL;DR

This paper establishes a strong inverse theorem for Gowers norms of trace functions over prime fields, providing new estimates and insights into their uniformity properties.

Contribution

It introduces a novel inverse theorem for Gowers norms specifically for trace functions of certain $ extit{ extlangle} extit{ extellipsis} extgreater$-adic sheaves over prime fields.

Findings

Proves estimates for Gowers uniformity norms of trace functions.

Establishes a strong inverse theorem for these functions.

Enhances understanding of the structure of trace functions in additive combinatorics.

Abstract

We prove estimates for the Gowers uniformity norms of functions over $\Zz/p\Zz$ which are trace functions of certain $\ell$-adic sheaves, and establish in particular a strong inverse theorem for these functions.