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

Given a real number τ, we study the approximation of τ by signed harmonic sums σN(τ):=∑n≤Nsn(τ)/n, where the sequence of signs (sN(τ))N∈N is defined “greedily” by setting sN+1(τ):=+1 if σN(τ)≤τ, and sN+1(τ):=−1 otherwise. More precisely, we compute the limit points and the decay rate of the sequence (σN(τ)−τ)N∈N. Moreover, we give an accurate description of the behavior of the sequence of signs (sN(τ))N∈N, highlighting a surprising connection with the Thue–Morse sequence.