Character sums over affine spaces and applications

Abstract

Given a finite field ${\mathbb{F}}_q$, a positive integer $n$ and an ${\mathbb{F}}_q$-affine space $\mathcal{A}\subseteq {\mathbb{F}}_{q^n}$, we provide a new bound on the sum ${\sum }_{a\in \mathcal{A}}\chi (a)$, where $\chi$ a multiplicative character of ${\mathbb{F}}_{q^n}$. We focus on the applicability of our estimate to results regarding the existence of special primitive elements in ${\mathbb{F}}_{q^n}$. In particular, we obtain substantial improvements on previous works.

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