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

2020/06/04 by Dartyge, Cécile, Mérai, László, Winterhof, Arne
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2006.02791

Abstract

Let q=p^r be the power of a prime p and (β₁,… ,β_r) be an ordered basis of \mathbbF_q over \mathbbF_p. For ξ=∑_j=1^r x_jβ_j∈ \mathbbF_q with digits x_j∈\mathbbF_p, we define the Rudin-Shapiro function R on \mathbbF_q by R(ξ)=∑_i=1^r-1 x_ix_i+1, ξ∈ \mathbbF_q. For a non-constant polynomial f(X)∈ \mathbbF_q[X] and c∈ \mathbbF_p we study the number of solutions ξ∈ \mathbbF_q 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.

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