Equidistribution of Primitive Normal Elements in Finite Fields

TL;DR

This work addresses the distribution of primitive normal elements in finite fields by employing finite Fourier transform techniques. It introduces both divisor-dependent and divisor-free indicator constructions to detect primitive and normal elements, and proves that the set of primitive normal elements forms a Salem set and is strongly equidistributed in $\mathbb{F}_{q^n}$. The analysis extends to analogous sets such as quadratic residues and primitive roots modulo large primes, establishing sharp FFT bounds that yield equidistribution. The results rely on precise character-sum estimates, Dedekind-domain totients, and a careful decomposition of the finite field to control exponential sums, providing a Fourier-analytic pathway to equidistribution in finite field contexts.

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 $p\geq 3$.

Theorems & Definitions (40)

• Theorem 1.1
• Lemma 2.1
• proof : Proof
• Lemma 2.2
• proof : Proof
• Definition 3.1
• Lemma 3.1
• Lemma 4.1
• Corollary 4.1
• proof