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Claim. The answer to Problem 527 is yes. Marcus Michelen and Mehtaab Sawhney, Convergent points for random power series on the unit circle, arXiv:2509.02729, submitted 2 September 2025 (14 pages, Creative Commons Attribution 4.0; library card), prove in Theorem 1.1 that if $a_n$ are complex numbers with $\sqrt n\,\lvert a_n\rvert\to0$ and $\epsilon_n$ are independent uniform signs, then almost surely there is a point $z$ with $\lvert z\rvert=1$ at which $\sum_n\epsilon_na_nz^n$ converges. Theorem 1.2 strengthens this: the set of such $z$ almost surely has Hausdorff dimension $1$. The question's hypotheses, real $a_n$ with $\sum\lvert a_n\rvert^2=\infty$ and $\lvert a_n\rvert=o(1/\sqrt n)$, are a special case of the theorem's, which needs neither the divergence of $\sum\lvert a_n\rvert^2$ nor any monotonicity of $\lvert a_n\rvert$; so the theorem answers the question as the site states it, and also under the stronger reading with $\lvert a_{n+1}\rvert\le\lvert a_n\rvert$ that the site raises as a possible intention of Erdős. The proof works along a sparse sequence of scales $N_1<N_2<\cdots$ whose ratios $N_{i+1}/N_i$ tend to infinity, between $(\log N_i)^{\omega(1)}$ and $N_i^{o(1)}$, controls the partial sums at each scale, replaces the random signs by Gaussian coefficients through a Lindeberg comparison and applies the Gaussian correlation inequality. The theorem statements follow the card's digest.

Acceptance. Reviewed: the site's curator, Thomas Bloom, labels the problem PROVED and credits this paper on erdosproblems.com/527, with the community database in agreement (proved since 8 September 2025). The paper is a preprint: its arXiv record lists no journal reference so refereed is not listed. No independent review is recorded in this repository and none is claimed.

Depends on. Nothing in this wiki: the argument is the paper's own.