On the distribution of the Rudin-Shapiro function for finite fields

Abstract

Let q = p r be the power of a prime p and (β1, . . . , βr) be an ordered basis of Fq over Fp. For ξ = r j=1 xjβj ∈ Fq with digits xj ∈ Fp, we define the Rudin-Shapiro function R on Fq byFor a non-constant polynomial f (X) ∈ Fq[X] and c ∈ Fp we study the number of solutions ξ ∈ Fq of R(f (ξ)) = c. If the degree d of f (X) is fixed, r ≥ 6 and p → ∞, the number of solutions is asymptotically p r−1 for any c. The proof is based on the Hooley-Katz Theorem.