Superpolynomial convergence in the Riemann Rearrangement Theorem

Let $x\in \mathbb{R}$ be arbitrary and consider the `greedy’ approximation of $x$ by signed harmonic sums: given $a_n={\sum }_{k\le n}{\epsilon }_k\mathrm{/}k$ with ${\epsilon }_k\in \{-1,1\}$, we set ${\epsilon }_{n+1}=1$ if $a_n\le x$ and ${\epsilon }_{n+1}=-1$ otherwise. Bettin-Molteni-Sanna showed (Adv. Math. 2020) that this procedure has remarkable approximation properties: for almost all $x\in \mathbb{R}$ one has superpolynomial convergence in the sense that for every $k\in \mathbb{N}$ there are infinitely many $n\in \mathbb{N}$ with $\mathrm{\mid }{a}_n-x\mathrm{\mid }\le n^{-k}$. We extend this result from $\pm 1\pm 1\mathrm{/}2\pm 1\mathrm{/}3\cdots \pm 1\mathrm{/}n$ to moment sequences, i.e. sequences defined as the moments of a measure $\mu$ supported on $[0,1]$.