Generic power series on subsets of the unit disk

TL;DR

This paper investigates the boundary behavior of generic power series with bounded coefficients, showing that on certain open sets within the unit disk, the image is dense in the complex plane, and characterizes coefficient sets with this property.

Contribution

It establishes the density of images of generic power series on specific open subsets of the unit disk and characterizes coefficient sets that exhibit this behavior.

Findings

For open sets with a non-real boundary point, the image of the power series is dense in .

The result does not hold for all open sets approaching the boundary.

Provides a characterization of coefficient sets that ensure the density property.

Abstract

In this paper, we examine the boundary behaviour of the generic power series $f$ with coefficients chosen from a fixed bounded set $\Lambda$ in the sense of Baire category. Notably, we prove that for any open subset $U$ of the unit disk $D$ with a non-real boundary point on the unit circle, $f(U)$ is a dense set of $\CC$. 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.