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| Format: | Preprint |
| Published: |
2025
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| Online Access: | https://arxiv.org/abs/2508.02489 |
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| _version_ | 1866911090728763392 |
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| author | Steinerberger, Stefan |
| author_facet | Steinerberger, Stefan |
| contents | 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 $μ$ supported on $[0,1]$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_02489 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Superpolynomial convergence in the Riemann Rearrangement Theorem Steinerberger, Stefan Dynamical Systems Classical Analysis and ODEs 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 $μ$ supported on $[0,1]$. |
| title | Superpolynomial convergence in the Riemann Rearrangement Theorem |
| topic | Dynamical Systems Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2508.02489 |