A note on the zeroes of the Fredholm series (Q2039116)

The author studies the following function analytic in the unit disc \(\mathbb{D}\): \(f(z) = \sum_{k=0}^\infty z^{2^n}\) (the corresponding Taylor series is often called the Fredholm series). \textit{D. Masser} [in: Diophantine approximation. Lectures given at the C.I.M.E. summer school, Cetraro, Italy, June 2000. Berlin: Springer. 1--51 (2003: Zbl 1049.11081)] posed the question whether this function has infinitely many zeroes in \(\mathbb{D}\). The author proves even a more general statement: for every \(a \in \mathbb{C}\) and every disk \(U_{z_0, r} = \{ z : |z -z_0| < r \}\) such that \(|z_0|=1\), \(r>0\), the equation \(f(z) =a\) has infinitely many roots in \(U_{z_0, r}\).