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

Abstract Let $$\varvec{F}_q$$ F q be the finite field of q elements, where $$q=p^r$$ = p r is a power prime p , and $$\left( \beta _1, _2, \dots _r \right) $$ β 1 , 2 ⋯ an ordered basis over $$\varvec{F}_p$$ . For $$\begin{aligned} \xi =\sum _{i=1}^rx_i\beta _i, \quad x_i\in \varvec{F}_p, \end{aligned}$$ ξ ∑ i x ∈ we define Thue–Morse or sum-of-digits function $$T(\xi )$$ T ( ) on by T(\xi )=\sum _{i=1}^{r}x_i. . given pattern length s with $$1\le s\le q$$ ≤ s vector $$\varvec{\alpha }=(\alpha _1,\ldots ,\alpha _s)\in \varvec{F}_q^s$$ α … different coordinates $$\alpha _{j_1}\not = \alpha _{j_2}$$ j ≠ j_1<j_2\le s$$ < polynomial $$f(X)\in \varvec{F}_q[X]$$ f X [ ] degree d $$\mathbf{c} =(c_1,\ldots ,c_s)\in \varvec{F}_p^s$$ c put \mathcal{T}(\mathbf{c} ,\varvec{\alpha },f)=\{\xi \in \varvec{F}_q : T(f(\xi +\alpha _i))=c_i,~i=1,\ldots ,s\}. { : + } In this paper will see that under some natural conditions, size $$\mathcal{T}(\mathbf{c} },f)$$ asymptotically same for all }$$ in both cases, $$p\rightarrow \infty → ∞ $$r\rightarrow respectively. More precisely, have \left||\mathcal{T}(\mathbf{c} \varvec{\alpha }, f) |- p^{r-s} \right|\le (d-1)q^{1/2} | - d / certain conditions monomials large improve bound as well find which not true. particular, if d<p$$ dichotomy valid $$s\le d$$ $$s\ge d+1$$ ≥ there are vectors },f)=\emptyset ∅ so fails sufficiently r The case $$s=1$$ was studied before Dartyge Sárközy.