Problem 527

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claims/: The 1 claim page of Problem 527, one per claimant's result; the problem's standing derives from them.

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Statement. Let $a_n\in \mathbb{R}$ be such that $\sum_n \lvert a_n\rvert^2=\infty$ and $\lvert a_n\rvert=o(1/\sqrt{n})$. Is it true that, for almost all $\epsilon_n=\pm 1$, there exists some $z$ with $\lvert z\rvert=1$ (depending on the choice of signs) such that

$$\sum_n \epsilon_n a_n z^n$$

converges?

Status. Proved. The site labels the problem PROVED and credits Michelen and Sawhney [MiSa25], whose Theorem 1.1 gives a convergent point almost surely and whose Theorem 1.2 gives a set of convergent points of Hausdorff dimension one. The accepted claim page Michelen and Sawhney 2025 records the result, on the site's acceptance; the paper is a preprint, so no refereed publication is listed. The frontmatter standing derives from the claim page.

Source. erdosproblems.com/527, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #527, https://www.erdosproblems.com/527.

References.

• [DE59] Dvoretzky, A. and Erdős, P., Divergence of random power series. Michigan Math. J. 6 (1959), 343-347.
• [MiSa25] M. Michelen and M. Sawhney, Convergent points for random power series on the unit circle. arXiv:2509.02729 (2025).

Formalization. None recorded.

Current assessment

The site's formulation of 2026-09-04 asks whether, for real coefficients with $\sum\lvert a_n\rvert^2=\infty$ and $\lvert a_n\rvert=o(1/\sqrt n)$, almost every choice of signs leaves some point of the unit circle at which the series converges. The site adds that it is unclear whether Erdős also meant to assume $\lvert a_{n+1}\rvert\le\lvert a_n\rvert$. The answer is yes under either reading: Theorem 1.1 of Michelen and Sawhney [MiSa25] needs only $\sqrt n\,\lvert a_n\rvert\to0$, for complex coefficients, and neither the divergence of $\sum\lvert a_n\rvert^2$ nor monotonicity, so the uncertain hypothesis does not affect the outcome. The result is accepted here on the site's credit and recorded on the claim page Michelen and Sawhney 2025; the arXiv record lists no journal reference so no refereed publication is listed. The theorem statements of [MiSa25] and [DE59] are those recorded on their library cards; no independent review of the proof is recorded. The status search covered the site, its forum thread, the arXiv record and the community database on 2026-10-07; the thread holds no proof claim, and no other claim of the result was found.

Known Results

• For any coefficients with $\sum\lvert a_n\rvert^2=\infty$, almost every choice of signs makes the series diverge at almost every point of the unit circle, a classical fact the site records as well known. The question is about the remaining measure-zero set of points.
• Dvoretzky and Erdős [DE59] (library card) prove that if $\lvert a_n\rvert\ge c/\sqrt n$ for some $c>0$ and all large $n$, then almost every choice of signs makes the series diverge at every point of the unit circle. The question asks whether this threshold is sharp.
• Michelen and Sawhney [MiSa25] prove that it is: under $\lvert a_n\rvert=o(1/\sqrt n)$ a convergent point exists almost surely, and the set of convergent points almost surely has Hausdorff dimension $1$ while, when $\sum\lvert a_n\rvert^2=\infty$, still having Lebesgue measure zero. This is the accepted claim Michelen and Sawhney 2025.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• dvoretzky_1959_divergence_random_power_series
• dvoretzky_1959_divergence_random_power_series / corollary
• dvoretzky_1959_divergence_random_power_series / remark_4_1
• dvoretzky_1959_divergence_random_power_series / theorem
• michelen_2025_convergent_points_random_power_series_unit