标题： Riemann重排定理中的超多项式收敛 显示英文标题

标题： Superpolynomial convergence in the Riemann Rearrangement Theorem

摘要： 设$x \in \mathbb{R}$为任意，并考虑用带符号调和和对$x$进行“贪心”逼近：给定$a_n = \sum_{k \leq n} \varepsilon_k/k$满足$\varepsilon_k \in \left\{-1,1\right\}$，我们设定$\varepsilon_{n+1} = 1$如果$a_n \leq x$且$\varepsilon_{n+1} = -1$否则。 Bettin-Molteni-Sanna 表明（Adv. Math. 2020) 表明该过程具有显著的逼近性质：对于几乎所有$x \in \mathbb{R}$，在某种意义上具有超多项式收敛性，即对于每个$k \in \mathbb{N}$，存在无限多个$n \in \mathbb{N}$使得$|a_n - x| \leq n^{-k}$。 我们将这一结果从$\pm 1 \pm 1/2 \pm 1/3 \dots \pm 1/n$扩展到矩序列，即定义为测度$\mu$在$[0,1]$上的矩的序列。 显示英文摘要

摘要： 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 $[0,1]$.