Construction of some Chowla sequences

Abstarct In this paper, we show that for a twice differentiable function \(g\) having countable zeros and for Lebesgue almost every \(\beta > 1\), the sequence \((e^{2\pi i \beta ^ng(\beta )})_{n\in {\mathbb {N}}}\) is orthogonal to all topological dynamical systems of zero entropy. To this end, we define the Chowla property and the Sarnak property for numerical sequences taking values 0 or complex numbers of modulus 1. We prove that the Chowla property implies the Sarnak property and show that for Lebesgue almost every \(\beta > 1\), the sequence \((e^{2\pi i \beta ^n})_{n\in {\mathbb {N}}}\) shares the Chowla property. It is also discussed whether the samples of a given random sequence have the Chowla property almost surely. Some dependent random sequences having almost surely the Chowla property are constructed