Superpolynomial convergence in the Riemann Rearrangement Theorem

Let x∈R be arbitrary and consider the ‘greedy’ approximation of x by signed harmonic sums: given an=∑k≤nεk/k with εk∈{−1,1}, we set εn+1=1 if an≤x and εn+1=−1 otherwise. Bettin–Molteni–Sanna showed [Adv. Math. 336 (2020)] that this procedure has remarkable approximation properties: for almost all x∈R one has superpolynomial convergence in the sense that for every k∈N there are infinitely many n∈N with |an−x|≤n−k. We extend this result from ±1±1/2±…±1/n to moment sequences, i.e. sequences defined as the moments of a measure μ supported on [0,1].