On arithmetic progressions in finite fields

Abstract

In this paper, we explore the existence of $m$-terms arithmetic progressions in ${\mathbb{F}}_{q^n}$ with a given common difference whose terms are all primitive elements, and at least one of them is normal. We obtain asymptotic results for $m\ge 4$ and concrete results for $m\in \{2,3\}$, where the complete list of exceptions when the common difference belongs to ${\mathbb{F}}_q^{\ast }$ is obtained. The proofs combine character sums, sieve estimates, and computational arguments using the software SageMath.