Double Sums of Kloosterman Sums in Finite Fields

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.