On some lacunary power series
The author continues the investigations of \textit{R. Salem} and \textit{A. Zygmund} [Duke Math. J. 12, 569--578 (1945; Zbl 0060.20402), \textit{A. S. Belov} [Mat. Sb. 181, No.~8, 1048--1060 (1990; Zbl 0723.30037)], \textit{A. Canton, A. Granados}, and \textit{Ch. Pommerenke} [Mich. Math. J. 52, No.~2, 279--287 (2004; Zbl 1067.30071)]. He proves the following. Theorem. There exists \(q \in (0, 1)\) with the following property. Let \(b = 2, 3, \dots\) and let \[ f (z) = \sum \limits_{n=0}^{\infty} a_n z^{b^n}, \] where \(a_n \in \mathbb C,\) such that \(\sum \limits_{n=0}^{\infty} | a_n| < \infty\) and \(| a_{n+1}| > q | a_n| \) for sufficiently large \(n.\) Then there exists a disk \(D \subset \mathbb C\) such that \(f\) assumes every value from \(D\) uncountable many times on the unit circle. Hence, \(f\) does not preserve Borel set on the unit circle.
scientific article; zbMATH DE number 5140511