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

2023/12/12 by Michael James Coons, Coons, Michael, Yohei Tachiya +1
Computer Science · Mathematics · #11A63 #11B85 #11J72 #Advanced Combinatorial Mathematics #Combinatorics (math.CO) #FOS: Computer and information sciences #FOS: Mathematics #Formal Languages and Automata Theory (cs.FL) #Mathematical Dynamics and Fractals #Number Theory (math.NT) #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.2312.06981

openalex publication_date 2023/12/12 · openalex created_date 2023/12/14 · openalex updated_date 2026/07/28

Abstract

The Thue--Morse sequence \t(n)\_{n\geqslant 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 β>√φ=1.272019649…, the set of the numbers 1, ∑_{n\geqslant 1}\fract(n)βⁿ, ∑_{n\geqslant 1}\fract(n²)βⁿ, …, ∑_{n\geqslant 1}\fract(n^k)βⁿ, … is linearly independent over the field ℚ(β), where φ:=(1+√(5))/2 is the golden ratio. Our result implies that for any k\geqslant 1 and for any a₁,a₂,…,a_k∈ℚ(β), not all zero, the sequence \a₁t(n)+a₂t(n²)+⋯+a_kt(n^k)\_{n\geqslant 1} cannot be eventually periodic.

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