Linear independence of series related to the Thue--Morse sequence along powers

Abstract

The Thue--Morse sequence $\{t(n){\}}_{n⩾1}$ is the indicator function of the parity of the number of ones in the binary expansion of positive 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 ${\hat{I}}^2>\sqrt{\ddot{I}}†=1.272019649\dots$, the set of the numbers $$ 1,\quad \sum_{n\geqslant 1}\frac{t(n)}{β^{n}},\quad \sum_{n\geqslant 1}\frac{t(n^2)}{β^{n}},\quad \dots, \quad \sum_{n\geqslant 1}\frac{t(n^k)}{β^{n}},\quad \dots $$ is linearly independent over the field $\mathbb{Q}({\hat{I}}^2)$, where $\ddot{I}†:=(1+\sqrt{5})/2$ is the golden ratio. Our result implies that for any $k⩾1$ and for any $a_1,a_2,\dots ,a_k\in \mathbb{Q}({\hat{I}}^2)$, not all zero, the sequence \{$a_{1}t(n)+a_{2}t(n^2)+\cdots +a_{k}t(n^k){\}}_{n⩾1}$ cannot be eventually periodic.

9 pages