Convergent points for random power series on the unit circle
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915476972503040 |
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| author | Michelen, Marcus Sawhney, Mehtaab |
| author_facet | Michelen, Marcus Sawhney, Mehtaab |
| contents | Consider a random power series of the form $P(z) = \sum_{n\ge 1} \varepsilon_n a_n z^{n}$ where $a_n \in \mathbb{C}$ are deterministic and $\varepsilon_n$ are chosen independently and uniformly at random from $\{\pm 1\}$. Kolmogorov's three-series theorem states that if $\sum_{n} |a_n|^2 = \infty$ then $P(z)$ almost-surely diverges at almost every $z$ with $|z| = 1$. Dvoretzky and Erdős proved in 1959 that if $|a_n| = Ω(1/\sqrt{n})$ then in fact $P$ almost surely diverges at every $|z| = 1$. Erdős then asked in 1961 if this is sharp, meaning that if $|a_n| = o(1/\sqrt{n})$ then there is almost surely some convergent point $z$ with $|z| = 1$. We prove this in a strong sense and show that if $a_n = o(1/\sqrt{n})$ then in fact the set of convergent points of $P$ with $|z| = 1$ has Hausdorff dimension $1$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_02729 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Convergent points for random power series on the unit circle Michelen, Marcus Sawhney, Mehtaab Probability Classical Analysis and ODEs Consider a random power series of the form $P(z) = \sum_{n\ge 1} \varepsilon_n a_n z^{n}$ where $a_n \in \mathbb{C}$ are deterministic and $\varepsilon_n$ are chosen independently and uniformly at random from $\{\pm 1\}$. Kolmogorov's three-series theorem states that if $\sum_{n} |a_n|^2 = \infty$ then $P(z)$ almost-surely diverges at almost every $z$ with $|z| = 1$. Dvoretzky and Erdős proved in 1959 that if $|a_n| = Ω(1/\sqrt{n})$ then in fact $P$ almost surely diverges at every $|z| = 1$. Erdős then asked in 1961 if this is sharp, meaning that if $|a_n| = o(1/\sqrt{n})$ then there is almost surely some convergent point $z$ with $|z| = 1$. We prove this in a strong sense and show that if $a_n = o(1/\sqrt{n})$ then in fact the set of convergent points of $P$ with $|z| = 1$ has Hausdorff dimension $1$. |
| title | Convergent points for random power series on the unit circle |
| topic | Probability Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2509.02729 |