Suppose f(z) is a power series with positive integer coefficients centered at zero and positive radius of convergence. What is the likelihood that f has a dense set of singularities on its circle of convergence (or any larger circle for that matter).

For instance, a classical theorem shown in Polya & Szego assumes f has integer coefficients and radius of convergence one. In this case, f is either rational or has a dense set of singularities on the unit circle. There are only countably many of the former and uncountably many of the latter, so the latter case is ubiquitous. (The supplied proof depends heavily on integer coefficients and radius 1.)

What can be said for more general radii? This question arises from combinatorial investigations: such f are generating functions for counting problems and several that I have run into seem to have dense sets of singularities on a circle (they also have a dominant real singularity, per Pringsheim's theorem, with slightly smaller modulus).