A Residual Class of Holomorphic Functions

In their 1975 landmark paper, D. D. Bonar and F. W. Carroll have shown that, in the sense of category, there exists a residual class SA of "strongly annular" holomorphic functions in the open unit disk D such that, for each f in SA, there exists an open subset O-f,O-infinity of D such that (1) O-f,O-infinity contains a sequence of concentric circles of increasing radii converging to the unit circle and (2) f(z) -> infinity as vertical bar z vertical bar -> 1 through O-f,O-infinity. Because circles have 2-dimensional Lebesgue measure zero, it has been an open question as to whether the set O-f,O-infinity could be chosen to have 2-dimensional measure-theoretic thickness. Here we give a definitive answer to that question. We show that for most functions f in SA, the set O-f,O-infinity can be chosen so that it has upper global metric density 1. Even more, we show that for most functions f in SA and for every complex value omega there exists an open subset O-f,O-infinity of D that has upper global metric density 1 such that f(z) converges to omega as vertical bar z vertical bar -> 1 through O-f,O-omega.