The Thue–Morse sequence $\{t(n)\}_{n\geqslant 0}$ is the indicator function of the parity of the number of ones in the binary expansion of nonnegative integers n, where $t(n)=1$ (resp. $=0$) if the binary expansion of n has an odd (resp. even) number of ones. In this paper, we generalize a recent result of E. Miyanohara by showing that, for a fixed Pisot or Salem number $\beta>\sqrt {\varphi }=1.272019\ldots $, the set of the numbers $$\begin{align*}1,\quad \sum_{n\geqslant1}\frac{t(n)}{\beta^{n}},\quad \sum_{n\geqslant1}\frac{t(n^2)}{\beta^{n}},\quad \dots, \quad \sum_{n\geqslant1}\frac{t(n^k)}{\beta^{n}},\quad \dots \end{align*}$$is linearly independent over the field $\mathbb {Q}(\beta )$, where $\varphi :=(1+\sqrt {5})/2$ is the golden ratio. Our result yields that for any integer $k\geqslant 1$ and for any $a_1,a_2,\ldots ,a_k\in \mathbb {Q}(\beta )$, not all zero, the sequence {$a_1t(n)+a_2t(n^2)+\cdots +a_kt(n^k)\}_{n\geqslant 1}$ cannot be eventually periodic.

Thue-Morse 序列$\{t(n)\}_{n\geqslant 0}$是非负整数n的二进制展开式中 1 数量奇偶校验的指示函数，其中$t(n)=如果n的二进制展开式有奇数（或偶数）个数，则为1$（或$=0$ ）。在本文中，我们概括了 E. Miyanohara 的最新结果，表明对于固定的 Pisot 或 Salem 数$\beta>\sqrt {\varphi }=1.272019\ldots $，数字$$\begin{的集合对齐*}1,\quad \sum_{n\geqslant1}\frac{t(n)}{\beta^{n}},\quad \sum_{n\geqslant1}\frac{t(n^2)} {\beta^{n}},\quad \dots, \quad \sum_{n\geqslant1}\frac{t(n^k)}{\beta^{n}},\quad \dots \end{align *}$$在域$\mathbb {Q}(\beta )$上线性无关，其中$\varphi :=(1+\sqrt {5})/2$是黄金比例。我们的结果得出，对于任何整数$k\geqslant 1$和任何$a_1,a_2,\ldots ,a_k\in \mathbb {Q}(\beta )$，不全为零，序列 { $a_1t(n) +a_2t(n^2)+\cdots +a_kt(n^k)\}_{n\geqslant 1}$最终不可能是周期性的。