Construction of some Chowla sequences

Abstract

For numerical sequences taking values $0$ or complex numbers of modulus $1$, we define Chowla property and Sarnak property. We prove that Chowla property implies Sarnak property. We also prove that for Lebesgue almost every $\beta>1$, the sequence $(e^{2\pi \beta^n})_{n\in \mathbb{N}}$ shares Chowla property and consequently is orthogonal to all topological dynamical systems of zero entropy. It is also discussed whether the samples of a given random sequence have Chowla property almost surely. Some dependent random sequences having almost surely Chowla property are constructed.