Rubel's problem on bounded analytic functions

Abstract

The paper shows that for any $G_{\delta }$ set $F$ of Lebesgue measure zero on the unit circle $T$ there exists a function $f\in H^{\infty }$ such that the radial limits of $f$ exist at each point of $T$ and vanish precisely on $F$. This solves a problem proposed by Lee Rubel in 1973.