The average density of K-normal elements over finite fields

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 $\{\alpha ,{\alpha }^q,\dots ,{\alpha }^{q^{n-1}}\}$ 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 $$\underset{t\to +\infty }{\lim }\frac{1}{t}\sum _{1\le n\le t}{\lambda }_{q,k}(n),$$ exists and it is positive.

arXiv comment8 pages; comments are welcome