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

The Thue–Morse sequence (n) 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 β>√φ =1.272019… , the set of the numbers align*1, ∑ₙ₁t(n)βⁿ, ∑ₙ₁t(n²)βⁿ, , ∑ₙ₁t(nᵏ)βⁿ, align* is linearly independent over the field Q(β ) , where φ :=(1+√5)/2 is the golden ratio. Our result yields that for any integer k 1 and for any a₁,a₂,… ,aₖ∈ Q(β ) , not all zero, the sequence { a₁t(n)+a₂t(n²)+⋯ +aₖt(nᵏ) 1 cannot be eventually periodic.