Convergent points for random power series on the unit circle

Abstract

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 Erd\H{o}s proved in 1959 that if $|a_n| = \Omega(1/\sqrt{n})$ then in fact $P$ almost surely diverges at every $|z| = 1$. Erd\H{o}s 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 $1$.