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) := Sigma(n <= N) s(n)(tau)/n, the sequence of signs (s(N)(tau))(N is an element of N) is defined "greedily" by setting s(N+1)(tau) := +1 if sigma(N)(tau) <= tau, and s(N+1)(tau) := -1 otherwise. More precisely, we compute the limit points and the decay rate of the sequence (sigma(N)(tau) - tau)(N is an element of N). Moreover, we give an accurate description of the behavior of the sequence of signs (s(N)(tau))(N is an element of N,) highlighting a surprising connection with the Thue-Morse sequence. (C) 2020 Elsevier Inc. All rights reserved.