Sufficient Conditions for Strong Natural Boundaries involving Lacunary Sequences

Abstract

The notion of a regular point is extended to that of a weak regular point, and so the notion of a strong natural boundary may be described locally as the absence of weak regular points, just as a natural boundary is locally the absence of regular points. The main result is a sufficient condition for a strong natural boundary, which is the conjunction of the unboundedness of the coefficients of the underlying power series with either of two specific gap density conditions imposed on the exponents that yields a lacunary sequence.