Superpolynomial convergence in the Riemann Rearrangement Theorem

TL;DR

This paper extends the superpolynomial convergence results of a greedy approximation method for real numbers, originally applied to harmonic sums, to more general moment sequences derived from measures on [0,1].

Contribution

It generalizes previous superpolynomial convergence results from harmonic sums to sequences of moments of measures on [0,1], broadening the scope of the approximation method.

Findings

Almost all real numbers are approximated with superpolynomial speed.

The approximation method applies to a wider class of sequences beyond harmonic sums.

Superpolynomial convergence is achieved for sequences derived from measures on [0,1].

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. 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 $\pm 1 \pm 1/2 \pm 1/3 \dots \pm 1/n$ to moment sequences, i.e. sequences defined as the moments of a measure $\mu$ supported on $[0,1]$.