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 \begin{equation*} \xi =\sum _{j=1}^r x_j\beta _j\in \mathbb {F}_q \quad \text {with digits }x_j\in \mathbb {F}_p, \end{equation*} we define the Rudin-Shapiro function $R$ on $\mathbb {F}_q$ by \begin{equation*} R(\xi )=\sum _{i=1}^{r-1} x_ix_{i+1}, \quad \xi \in \mathbb {F}_q. \end{equation*} 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.