# Double Sums of Kloosterman Sums in Finite Fields

## TL;DR

This paper establishes bounds for double sums of Kloosterman sums over finite fields, extending recent additive combinatorics techniques to analyze sums with parameters over affine spaces.

## Contribution

It introduces new bounds for double Kloosterman sums in finite fields using advanced additive combinatorics methods, generalizing previous results.

## Key findings

- Bounded double sums of Kloosterman sums over finite fields.
- Extended finite field analogues of recent residue ring results.
- Applied additive combinatorics to finite field exponential sums.

## Abstract

We bound double sums of Kloosterman sums over a finite field ${\mathbb F}_{q}$, with one or both parameters ranging over an affine space over its prime subfield ${\mathbb F}_p \subseteq {\mathbb F}_{q} $. These are finite fields analogues of a series of recent results by various authors in finite fields and residue rings. Our results are based on recent advances in additive combinatorics in arbitrary finite field.