Convergent points for random power series on the unit circle

Abstract

Consider a random power series of the form $P(z)={\sum }_{n\ge 1}{\epsilon }_n{a}_n{z}^n$ where $a_n\in \mathbb{C}$ are deterministic and ${\epsilon }_n$ 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 $|a_n|=\hat{I}\text{c◯}(1/\sqrt{n})$ 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 $1$.

14 pages