On the transcendental values of Cantor-like power series

Abstract

Recently, Laohakosol and Sripayap (East-West J. Math.19 (2017) 65-79) considered certain Cantor-like power series. They investigated the nature of such series evaluated at rational points. Using Roth's theorem, they proved that these series take transcendental values at rational points subject to certain conditions on the growth of the coefficients of these series. In the present paper, we consider these Cantor-like power series treated by Laohakosol and Sripayap at algebraic points and at Liouville number points. Using a theorem due to LeVeque, a lemma due to Icen, and Laohakosol and Sripayap's technique, we first prove that these Cantor-like power series take transcendental values at algebraic points. Using Roth's theorem and Laohakosol and Sripayap's technique, we then prove that these series take rational or transcendental values at Liouville number points.