Greedy approximations by signed harmonic sums and the Thue--Morse sequence

Given a real number $\tau$, we study the approximation of $\tau$ by signed harmonic sums ${\sigma }_N(\tau ):={\sum }_{n\le N}{s}_n(\tau )\mathrm{/}n$, where the sequence of signs $(s_N(\tau ){)}_{N\in \mathbb{N}}$ is defined “greedily” by setting $s_{N+1}(\tau ):=+1$ if ${\sigma }_N(\tau )\le \tau$, 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.