Greedy approximations by signed harmonic sums and the Thue-Morse sequence. (English) Zbl 1436.11080
In this interesting paper, the authors develop further their result of S. Bettin et al. [C. R., Math., Acad. Sci. Paris 356, No. 11–12, 1062–1074 (2018; Zbl 1434.11063)].
Given a real number \(\tau\), the authors study the approximation of \(\tau\) by signed harmonic sums \(\sigma_N(\tau))=\sum_{n\le N}s_n(\tau)/n\), where the sequence of signs \((s_N(\tau))_{N\in\mathbb{N}}\) is defined by \(s_{N+1}(\tau)=+1\) if \(\sigma_N(\tau)\le\tau\), and \(s_{N+1}(\tau)=-1\) otherwise. We call this a greedy approximation to \(\tau\) by signed harmonic sums.
Surprisingly, the behavior of the sequence \(s_N(\tau)\) is not chaotic, but is extremely structured and allows the authors to prove precise results on the asymptotic behavior of this sequence, also visualizing that behavior.
They compute the limit points and the decay rate of the sequence \((\sigma_N(\tau)-\tau)_{n\in\mathbb{N}}\). Accurately describing the behavior of the sequence of signs \(s_N(\tau)\), the authors point out a surprising connection with the Thue-Morse sequence. This result extends a result of J.-P. Allouche and H. Cohen [Bull. Lond. Math. Soc. 17, 531–538 (1985; Zbl 0577.10036)].
Given a real number \(\tau\), the authors study the approximation of \(\tau\) by signed harmonic sums \(\sigma_N(\tau))=\sum_{n\le N}s_n(\tau)/n\), where the sequence of signs \((s_N(\tau))_{N\in\mathbb{N}}\) is defined by \(s_{N+1}(\tau)=+1\) if \(\sigma_N(\tau)\le\tau\), and \(s_{N+1}(\tau)=-1\) otherwise. We call this a greedy approximation to \(\tau\) by signed harmonic sums.
Surprisingly, the behavior of the sequence \(s_N(\tau)\) is not chaotic, but is extremely structured and allows the authors to prove precise results on the asymptotic behavior of this sequence, also visualizing that behavior.
They compute the limit points and the decay rate of the sequence \((\sigma_N(\tau)-\tau)_{n\in\mathbb{N}}\). Accurately describing the behavior of the sequence of signs \(s_N(\tau)\), the authors point out a surprising connection with the Thue-Morse sequence. This result extends a result of J.-P. Allouche and H. Cohen [Bull. Lond. Math. Soc. 17, 531–538 (1985; Zbl 0577.10036)].
Reviewer: István Gaál (Debrecen)
References:
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