Problem 4 in Chapter 4 of Stein's book "Real Analysis" says $\sum_{n\geqslant 0}z^{2^n}$ doesn't have radial limit as $z$ approaches the unit circle from inside almost everywhere. It's fairly easy to find a dense set s.t. the radial limit doesn't exist ($=\infty$, actually), but is there a simple way to prove the set of divergence has positive measure?