Given a real number $\tau$, we study the approximation of $\tau$ by signed harmonic sums $\sigma_N(\tau) := \sum_{n \leq N}{s_n(\tau)}/n$, where the sequence of signs $(s_N(\tau))_{N \in\mathbb{N}}$ is defined "greedily" by setting $s_{N+1}(\tau) := +1$ if $\sigma_N(\tau) \leq \tau$, and $s_{N+1}(\tau) := -1$ otherwise. Precisely, we compute the limit points and the decay rate of the sequence $(\sigma_N(\tau)-\tau)_{N \in \mathbb{N}}$. Moreover, we give an accurate description of the behavior of the sequence of signs $(s_N(\tau))_{N\in\mathbb{N}}$, highlighting a surprising connection with the Thue--Morse sequence.

中文翻译：

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有符号调和和 Thue-Morse 序列的贪婪逼近

给定一个实数 $\tau$，我们通过有符号调和和 $\sigma_N(\tau) := \sum_{n \leq N}{s_n(\tau)}/n$ 研究 $\tau$ 的近似值，其中符号序列 $(s_N(\tau))_{N \in\mathbb{N}}$ 是通过设置 $s_{N+1}(\tau) := +1$ if $ 来“贪婪地”定义的\sigma_N(\tau) \leq \tau$ 和 $s_{N+1}(\tau) := -1$ 否则。准确地说，我们计算了序列 $(\sigma_N(\tau)-\tau)_{N \in \mathbb{N}}$ 的极限点和衰减率。此外，我们准确描述了符号序列 $(s_N(\tau))_{N\in\mathbb{N}}$ 的行为，突出了与 Thue--Morse 序列的惊人联系。