The distribution of singular points of the sum of a series of exponential monomials on the boundary of its domain of convergence

Abstract

Abstract The problem of the distribution of the singular points of the sum of a series of exponential monomials on the boundary of the domain of convergence of the series is considered. Sufficient conditions are found for a singular point to exist on a prescribed arc on the boundary; these are stated in purely geometric terms. The singular point exists due to simple relations between the maximum density of the exponents of the series in an angle and the length of the arc on the boundary of the domain of convergence that corresponds to this angle. Necessary conditions for a singular point to exist on a prescribed arc on the boundary are also obtained. They are stated in terms of the minimum density of the exponents in an angle and the length of the arc. On this basis, for sequences with density, criteria are established for the existence of a singular point on a prescribed arc on the boundary of the domain of convergence. Bibliography: 27 titles.