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

TL;DR

This paper investigates the distribution of the Thue-Morse function over finite fields along polynomial values, establishing asymptotic uniformity results and bounds under certain conditions, with improvements for large degree monomials.

Contribution

It provides new bounds and conditions for the normality of the Thue-Morse function on finite fields, extending previous results and identifying when uniform distribution fails.

Findings

Asymptotic size of certain sets matches expected value within bounds

Bounds are improved for monomials of large degree

Identifies conditions where uniformity does not hold for higher pattern lengths

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 $$\xi=\sum_{i=1}^rx_i\beta_i, \quad \quad x_i\in{\mathbb F}_p,$$ 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 ${\cal A}=\{\alpha_1,\ldots,\alpha_s\}\subset {\mathbb F}_q$, a polynomial $f(X)\in{\mathbb F}_q[X]$ of degree $d$ and a vector $\underline{c}=(c_1,\ldots,c_s)\in{\mathbb F}_p^s$ we put \[ {\cal T}(\underline{c},{\cal A},f)=\{\xi\in{\mathbb F}_q : T(f(\xi+\alpha_i))=c_i,~i=1,\ldots,s\}. \] In this paper we will see that under some natural conditions, the size of~${\cal T}(\underline{c},{\cal A},f)$ is asymptotically the same for all~$\underline{c}$ and ${\cal A}$ in both cases, $p\rightarrow \infty$ and $r\rightarrow \infty$, 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 ${\cal A}$ if $s\ge d+1$. The case $s=1$ was studied before by Dartyge and S\'ark\"ozy.