Linear independence of the real numbers generated by the square and cube subsequences of Thue–Morse

Let \((t(m))_{m \ge0}\) ${(t(m))}_{m\ge 0}$ be Thue-Morse sequence and \(b>2\) $b>2$ be an integer.In this paper, we prove that the real numbers \(1\) $1$ , \(\sum_{m=0}^\infty {\frac{t(m^2)}{{b}^{m+1}}}\) ${\sum }_{m=0}^{\infty }\frac{t(m^2)}{b^{m+1}}$ and \(\sum_{m=0}^\infty {\frac{t(m^3)}{{b}^{m+1}}}\) ${\sum }_{m=0}^{\infty }\frac{t(m^3)}{b^{m+1}}$ arelinearly independent over \(\mathbb{Q}\) $\mathbb{Q}$ .