Construction of some Chowla sequences

In this paper, we show that for a twice differentiable function g having countable zeros and for Lebesgue almost every $$\beta >1$$ β > 1 , the sequence $$(e^{2\pi i \beta ^ng(\beta )})_{n\in {\mathbb {N}}}$$ ( e 2 π i β n g ( β ) ) n ∈ 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 $$\beta >1$$ β > 1 , the sequence $$(e^{2\pi i \beta ^n})_{n\in {\mathbb {N}}}$$ ( e 2 π i β n ) n ∈ 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.