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

Abstract: Let $q=p^r$ be the power of a prime $p$ and $(\beta_1,\ldots ,\beta_r)$ be an ordered basis of $\mathbb{F}q$ over $\mathbb{F}_p$. For $$ \xi=\sum\limits{j=1}^r x_j\beta_j\in \mathbb{F}q \quad \mbox{with digits }x_j\in\mathbb{F}_p, $$ we define the Rudin-Shapiro function $R$ on $\mathbb{F}_q$ by $$ R(\xi)=\sum\limits{i=1}^{r-1} x_ix_{i+1}, \quad \xi\in \mathbb{F}_q. $$ 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 $p\rightarrow \infty$, the number of solutions is asymptotically $p^{r-1}$ for any $c$. The proof is based on the Hooley-Katz Theorem.