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

Abstract

Let $q=p^r$ be the power of a prime $p$ and $({\beta }_1,\dots ,{\beta }_r)$ be an ordered basis of ${\mathbb{F}}_q$ over ${\mathbb{F}}_p$. For $$ ξ=\sum\limits_{j=1}^r x_jβ_j\in \mathbb{F}_q \quad \mbox{with digits }x_j\in\mathbb{F}_p, $$ we define the Rudin-Shapiro function $R$ on ${\mathbb{F}}_q$ by $$R(\xi )=\sum _{i=1}^{r-1}{x}_i{x}_{i+1},\xi \in {\mathbb{F}}_q.$$ For a non-constant polynomial $f(X)\in {\mathbb{F}}_q[X]$ and $c\in {\mathbb{F}}_p$ we study the number of solutions $\xi \in {\mathbb{F}}_q$ of $R(f(\xi ))=c$. If the degree $d$ of $f(X)$ is fixed, $r\ge 6$ and $p\to \infty$, the number of solutions is asymptotically $p^{r-1}$ for any $c$. The proof is based on the Hooley-Katz Theorem.