On exponential sums over orbits in $\mathbb{F}_p^d$

TL;DR

This paper establishes bounds for exponential sums over vector orbits in finite fields using approximate group classification, advancing understanding of algebraic and combinatorial structures in finite field actions.

Contribution

It introduces new bounds for exponential sums over orbits in finite fields by applying a classification theorem for approximate groups.

Findings

Bound for exponential sums over orbits in $\,\mathbb{F}_p^d$

Application of approximate group classification theorem

Enhanced understanding of orbit structure in finite fields

Abstract

This paper proves a bound for exponential sums over orbits of vectors in $\mathbb{F}_p^d$ under subgroups of $\rm{GL}_d(\mathbb{F}_p)$. The main tool is a classification theorem for approximate groups due to Gill, Helfgott, Pyber, and Szab\'o.