Boundary behaviour of inner functions and holomorphic mappings

Let f be a holomorphic function in the unit disk omitting a set A of values of the complex plane. If A has positive logarithmic capacity, R. Nevanlinna proved that f has a radial limit at almost every point of the unit circle. If A is any infinite set, we show that f has a radial limit at every point of a set of Hausdorff dimension 1. A localization technique reduces this result to the following theorem on inner functions. If I is an inner function omitting a set of values B in the unit disk, then for any accumulation point b of B in the disk, there exists a set of Hausdorff dimension 1 of points in the circle where I has radial limit b.