Values of Cantor-like analytic functions at rational points

Abstract

Consider a Cantor-like analytic function with rational coefficients. The nature of such a function evaluated at rational points is investigated. First, using the Roth’s theorem, the transcendence of its values evaluated at rational points is derived subject to certain conditions on the growth of the coefficients. Second, using a technique of Mahler, we show that the values of a lacunary Cantor-like analytic function evaluated at algebraic points are algebraic if and only if their corresponding partial sums vanish