Superpolynomial convergence in the Riemann Rearrangement Theorem
Volume 290 / 2026
Abstract
Let $x \in \mathbb {R}$ be arbitrary and consider the ‘greedy’ approximation of $x$ by signed harmonic sums: given $a_n = \sum _{k \leq n} \varepsilon _k/k$ with $\varepsilon _k \in \left \{-1,1\right \}$, we set $\varepsilon _{n+1} = 1$ if $a_n \leq x$ and $\varepsilon _{n+1} = -1$ otherwise. Bettin–Molteni–Sanna showed [Adv. Math. 336 (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 $|a_n - x| \leq n^{-k}$. We extend this result from $\pm 1 \pm 1/2 \pm \dots \pm 1/n$ to moment sequences, i.e. sequences defined as the moments of a measure $\mu $ supported on $[0,1]$.
