Generic power series on subsets of the unit disk

Abstract

We examine the boundary behaviour of the generic power series f with coefficients chosen from a fixed bounded set Λ in the sense of Baire category. Notably, we prove that for any open subset U of the unit disk D with a nonreal boundary point on the unit circle, f(U) is a dense set of ℂ. As it is demonstrated, this conclusion does not necessarily hold for arbitrary open sets accumulating to the unit circle. To complement these results, a characterization of coefficient sets having this property is given.