Michelen 2025 convergent points random power series unit

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Marcus Michelen, Mehtaab Sawhney, Convergent points for random power series on the unit circle. arXiv:2509.02729 (2025). The arXiv record (https://arxiv.org/abs/2509.02729, read 2026-10-02) names the Creative Commons Attribution 4.0 license.

For a random power series P(z) = sum eps_n a_n z^n with deterministic complex coefficients a_n and independent uniform signs eps_n, Dvoretzky and Erdős showed in 1959 that |a_n| = Omega(1/sqrt n) forces almost sure divergence at every point of the unit circle. Erdős asked in 1961 whether this is sharp, i.e. whether |a_n| = o(1/sqrt n) already forces the existence of some convergent point on |z| = 1. Theorem 1.1 confirms this: if n^{1/2}|a_n| -> 0 then almost surely there exists z with |z| = 1 at which the series converges. Theorem 1.2 strengthens it substantially, showing the set of convergent points on the circle almost surely has Hausdorff dimension 1 (while still having Lebesgue measure 0 when sum |a_n|^2 diverges). The proof runs a multi-scale branching argument over events controlling partial sums along a sparse sequence of scales N_1 < N_2 < ..., whose ratios N_{i+1}/N_i tend to infinity, lying between (log N_i)^omega(1) and N_i^o(1) (p. 2; display (2.1), p. 4). It replaces Rademacher by Gaussian coefficients via a Lindeberg argument and then applies the Gaussian correlation inequality. This settles Erdős's question recorded as Problem 527.

Source: https://arxiv.org/abs/2509.02729.

Bears on. #527

Results to transcribe.

• Theorem 1.1: If a_n are complex with n^{1/2}|a_n| -> 0 and eps_n are independent uniform signs, then almost surely some z with |z| = 1 has sum eps_n a_n z^n convergent.
• Theorem 1.2: Under the same hypothesis, the set of z on the unit circle at which the series converges almost surely has Hausdorff dimension 1.
• Lemma 2.3: On the complement of two null events E_1 and E_2 (Lemmas 2.1 and 2.2: large oscillations between the sparse indices r_{k,j}, and large differentiated partial sums, each for infinitely many k), P(e(theta)) converges for every theta in the set A of (2.4) of points staying within N_k^{-1}(log N_k)^{-5} of a point alive at step k for every k (alive points satisfy the paper's two-part scale-wise smallness condition (2.2)); the proof shows the partial sums form a Cauchy sequence.
• Lemma 2.5: Lindeberg-type comparison replacing the Rademacher coefficients by Gaussian ones so the Gaussian correlation inequality applies.