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

Let ${\mathbb{F}}_q$ be the finite field of $q$ elements, where $q=p^r$ is a power of the prime $p$, and $({\beta }_1,{\beta }_2,\dots ,{\beta }_r)$ 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(\xi)=\sum_{i=1}^{r}x_i.%,\quad \xi=x_1\beta_1+\cdots +x_r\beta_r\in {\mathbb F}_q. For a given pattern length $s$ with $1\le s\le q$, a subset ${\cal A}={\alpha_1,\ldots,\alpha_s}\subset {\mathbb F}_q$,apolynomial$f(X)\in{\mathbb F}_q[X]$ofdegree$d$andavector$\underline{c}=(c_1,\ldots,c_s)\in{\mathbb F}_p^s$ we put $$\mathcal{T}(\underset{‾}{c},\mathcal{A},f)=\{\xi \in {\mathbb{F}}_q:T(f(\xi +{\alpha }_i))=c_i,i=1,\dots ,s\}\mathrm{.}$$ In this paper we will see that under some natural conditions, the size of~$\mathcal{T}(\underset{‾}{c},\mathcal{A},f)$ is asymptotically the same for all~$\underset{‾}{c}$ and $\mathcal{A}$ in both cases, $p\to \mathrm{\infty }$ and $r\to \mathrm{\infty }$, respectively. More precisely, we have $$\mid \mathrm{\mid }\mathcal{T}(\underset{‾}{c},\mathcal{A},f)\mathrm{\mid }-p^{r-s}\mid \le (d-1)q^{1\mathrm{/}2}$$ 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 $\underset{‾}{c}$ and $\mathcal{A}$ if $s\ge d+1$. The case $s=1$ was studied before by Dartyge and S'ark"ozy.