Boundary properties of holomorphic functions in the ball of \({\mathbb{C}}^ n\) (Q1090811)

A theorem of \textit{F. Bagemihl} and \textit{W. Seidel} [Math. Z. 61, 186-199 (1954; Zbl 0058.061)] and \textit{W. Rudin} [Bull. Am. Math. Soc. 60, 545 (1954)], concerning radial cluster sets of analytic functions in the disc is generalized to the unit ball of \({\mathbb{C}}^ n.\) Theorem: There exists an \(F_{\sigma}\)-set E of total measure in \(\partial B\) with the property: for every continuous function \(\phi\) in B, there is a holomorphic function f in B such that \(\lim_{r\to 1}(f(r\zeta)-\phi (r\zeta))=0\) for every \(\zeta\in E.\) The theorem is extended in a note added in proof to the case of certain bounded, pseudoconvex, starshaped domains.