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. (Finite Fields Appl. 24:170-183, 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 notion of 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, for arbitrary q and k, the arithmetic function $\lambda _{q, k}(n)$ has positive mean value, in the sense that the limit

$$\begin{aligned} \lim \limits _{t\rightarrow +\infty }\frac{1}{t}\sum _{1\le n\le t}\lambda _{q, k}(n), \end{aligned}$$

exists and it is positive.