The average density of K-normal elements over finite fields.

Let q be a prime power and, for each positive integer n ≥ 1 , let 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 ≤ k ≤ n , an element α ∈ F q n is k-normal over F q if the F q -vector space generated by the elements in the set { α , α q , ... , α 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 λ q , n , k of elements α ∈ F q n that are k-normal over F q and the density λ q , k (n) = λ q , n , k q n of such elements in F q n . In this paper we prove that, for arbitrary q and k, the arithmetic function λ q , k (n) has positive mean value, in the sense that the limit lim t → + ∞ 1 t ∑ 1 ≤ n ≤ t λ q , k (n) , exists and it is positive. [ABSTRACT FROM AUTHOR]