LA Probability Forum

USC KAP 414

Consider a random power series of the form P(z) = \sum_{n} a_n \xi_n z^n where a_n are deterministic and \xi_n are chosen independently and uniformly at random from {-1,1}. Kolmogorov's three-series theorem states that if the sum of the squares of |a_n| diverges then the random power series P(z) almost surely diverges at Lebesgue almost every z with |z| = 1. Seeking to understand when P in fact simultaneously diverges for *all* |z| = 1, Dvoretzky and Erdős proved in 1959 that if |a_n| = \Omega(1/sqrt(n)) then in fact P almost surely diverges at every z with |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 and in fact show that the set of convergent points has Hausdorff dimension 1. We will provide a gentle description of how one can think about random functions and outline how one may prove this theorem by thinking of the problem in terms of a branching process. No knowledge of complex analysis (or the definition of Hausdorff dimension) will be assumed. This is based on joint work with Mehtaab Sawhney.