Rubel's problem on bounded analytic functions

TL;DR

This paper solves Rubel's 1973 problem by constructing bounded analytic functions with radial limits vanishing exactly on any given Lebesgue measure zero G_delta set on the unit circle.

Contribution

It provides a method to realize any measure zero G_delta set as the zero set of radial limits of a bounded analytic function, answering a long-standing open problem.

Findings

Existence of bounded analytic functions with prescribed zero sets on the boundary

Construction method for functions with specific boundary behavior

Resolution of Rubel's problem from 1973

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.