Escaping Chaos in Random Multiplicative Functions
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arXiv
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866911709894017024 |
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| author | Xu, Max Wenqiang |
| author_facet | Xu, Max Wenqiang |
| contents | Let $f(n)$ be a Steinhaus random multiplicative function. Let $A\subset [1, N]$ be a finite set of integers. We show that \[\frac{1}{\sqrt{|A|}} \sum_{n\in A} f(n) \xrightarrow[]{d} \mathcal{CN}(0,1)\] forces that $|A|=o(N)$. We prove that the $o(1)$ density is sharp by showing that for most sets $A$, and thus confirm the existence, with density $ρ$ such that $(1-ρ)^{-1} =o((\log \log N)^{1/2})$, we have
\[ \frac{1}{\sqrt{(1-ρ) |A|}} \sum_{n\in A} f(n) \xrightarrow{d} \mathcal{CN}(0,1). \] The extra factor $\sqrt{1-ρ}$ makes a difference as long as the density $ρ>0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_21737 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Escaping Chaos in Random Multiplicative Functions Xu, Max Wenqiang Number Theory Probability Let $f(n)$ be a Steinhaus random multiplicative function. Let $A\subset [1, N]$ be a finite set of integers. We show that \[\frac{1}{\sqrt{|A|}} \sum_{n\in A} f(n) \xrightarrow[]{d} \mathcal{CN}(0,1)\] forces that $|A|=o(N)$. We prove that the $o(1)$ density is sharp by showing that for most sets $A$, and thus confirm the existence, with density $ρ$ such that $(1-ρ)^{-1} =o((\log \log N)^{1/2})$, we have \[ \frac{1}{\sqrt{(1-ρ) |A|}} \sum_{n\in A} f(n) \xrightarrow{d} \mathcal{CN}(0,1). \] The extra factor $\sqrt{1-ρ}$ makes a difference as long as the density $ρ>0$. |
| title | Escaping Chaos in Random Multiplicative Functions |
| topic | Number Theory Probability |
| url | https://arxiv.org/abs/2605.21737 |