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.
Categories: math.NT