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

Abstract

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