Equidistribution of Primitive Normal Elements in Finite Fields

TL;DR

This paper proves that primitive normal elements in finite fields are strongly equidistributed and form a Salem set, with similar results for quadratic residues and primitive roots modulo large primes.

Contribution

It establishes the equidistribution and Salem set properties of primitive normal elements in finite fields, extending to quadratic residues and primitive roots.

Findings

Primitive normal elements form a Salem set.

Primitive normal elements are strongly equidistributed.

Results also apply to quadratic residues and primitive roots.

Abstract

Let $q=p^k$ be a prime power, let $n\geq2$ be an integer and let $\mathbb{F}_{q^n}$ be a finite field. It is shown that the set of primitive normal elements is a Salem set. Furthermore, it is proved that this set is strongly equidistributed in the finite field. Similar results are proved for the set of quadratic residues and the set of primitive roots modulo a large prime $p\geq 3$.