Normality of the Thue-Morse function for finite fields along polynomial values

Abstract

Let Fq be the finite field of q elements, where q = p r is a power of the prime p, and (β1, β2, . . . , βr) be an ordered basis of Fq over Fp. Forwe define the Thue-Morse or sum-of-digits function T (ξ) on Fq byxi.For a given pattern length s with 1 ≤ s ≤ q, a subset A = {α1, . . . , αs} ⊂ Fq, a polynomial f (X) ∈ Fq[X] of degree d and a vector c = (c1, . . . , cs) ∈ F s p we putIn this paper we will see that under some natural conditions, the size of T (c, A, f ) is asymptotically the same for all c and A in both cases, p → ∞ and r → ∞, respectively. More precisely, we haveunder certain conditions on d, q and s. For monomials of large degree we improve this bound as well as we find conditions on d, q and s for which this bound is not true. In particular, if 1 ≤ d < p we have the dichotomy that the bound is valid if s ≤ d and fails for some c and A if s ≥ d + 1. The case s = 1 was studied before by Dartyge and Sárközy.