The average density of K-normal elements over finite fields

TL;DR

This paper studies the density of $k$-normal elements over finite fields, proving that their average density across field extensions has a positive limit, extending the understanding of normal element distributions.

Contribution

It establishes that the average density of $k$-normal elements over finite fields has a positive mean value, a new result in the distribution of these elements.

Findings

The mean value of the density of $k$-normal elements exists.

This mean value is strictly positive.

The result generalizes properties of normal elements to $k$-normal elements.

Abstract

Let $q$ be a prime power and, for each positive integer $n\ge 1$, let $\mathbb F_{q^n}$ be the finite field with $q^n$ elements. Motivated by the well known concept of normal elements over finite fields, Huczynska et al (2013) introduced the notion of $k$-normal elements. More precisely, for a given $0\le k\le n$, an element $\alpha\in \mathbb F_{q^n}$ is $k$-normal over $\mathbb F_q$ if the $\mathbb F_q$-vector space generated by the elements in the set $\{\alpha, \alpha^q, \ldots, \alpha^{q^{n-1}}\}$ has dimension $n-k$. The case $k=0$ recovers the normal elements. If $q$ and $k$ are fixed, one may consider the number $\lambda_{q, n, k}$ of elements $\alpha\in \mathbb F_{q^n}$ that are $k$-normal over $\mathbb F_q$ and the density $\lambda_{q, k}(n)=\frac{\lambda_{q, n, k}}{q^n}$ of such elements in $\mathbb F_{q^n}$. In this paper we prove that the arithmetic function $\lambda_{q, k}(n)$ has positive mean value, in the sense that the limit $$\lim\limits_{t\to +\infty}\frac{1}{t}\sum_{1\le n\le t}\lambda_{q, k}(n),$$ exists and it is positive.