Normality of the Thue-Morse function for finite fields along polynomial values

Abstract

Let ${\mathbb{F}}_q$ be the finite field of $q$ elements, where $q=p^r$ is a power of the prime $p$, and $\left({\beta }_1,{\beta }_2,\dots ,{\beta }_r\right)$ be an ordered basis of ${\mathbb{F}}_q$ over ${\mathbb{F}}_p$. For $$\xi =\sum _{i=1}^r{x}_i{\beta }_i,x_i\in {\mathbb{F}}_p,$$ we define the Thue-Morse or sum-of-digits function $T(\xi )$ on ${\mathbb{F}}_q$ by \[ T(ξ)=\sum_{i=1}^{r}x_i.%,\quad ξ=x_1β_1+\cdots +x_rβ_r\in {\mathbb F}_q. \] For a given pattern length $s$ with $1\le s\le q$, a subset $\mathcal{A}=\{{\alpha }_1,\dots ,{\alpha }_s\}\subset {\mathbb{F}}_q$, a polynomial $f(X)\in {\mathbb{F}}_q[X]$ of degree $d$ and a vector $\underline{c}=(c_1,\dots ,c_s)\in {\mathbb{F}}_p^s$ we put \[ {\cal T}(\underline{c},{\cal A},f)=\{ξ\in{\mathbb F}_q : T(f(ξ+α_i))=c_i,~i=1,\ldots,s\}. \] In this paper we will see that under some natural conditions, the size of~$\mathcal{T}(\underline{c},\mathcal{A},f)$ is asymptotically the same for all~$\underline{c}$ and $\mathcal{A}$ in both cases, $p\to \infty$ and $r\to \infty$, respectively. More precisely, we have \[ \left||{\cal T}(\underline{c},{\cal A},f)|-p^{r-s}\right|\le (d-1)q^{1/2}\] under certain conditions on $d,q$ and $s$. For monomials of large degree we improve this bound as well as we find conditions on $d,q$ and $s$ for which this bound is not true. In particular, if $1\le d<p$ we have the dichotomy that the bound is valid if $s\le d$ and fails for some $\underline{c}$ and $\mathcal{A}$ if $s\ge d+1$. The case $s=1$ was studied before by Dartyge and Sárközy.