The Chowla sequences and Sarnak sequences taking values in $\{-1,0,1\}$ were introduced by El Houcein el Abdalaoui, Joanna Kułaga-Przymus, Mariusz Lemańczyk and Thierry De La Rue for studying Chowla's conjecture and Sarnak's conjecture from ergodic theory point of view. One of their results is that Chowla sequences are always Sarnak sequences, and consequently orthogonal to all topological dynamical systems of zero entropy. In this talk, I will explain how to generalize Chowla sequences from $\{-1,0,1\}^\mathbb{N}$ to $(S^1\cup \{0\})^\mathbb{N}$. As an application, I will show that for almost all $\beta>1$, the sequences $(e^{2\pi i \beta^n})_{n\in \mathbb{N}}$ are Chowla sequences and consequently orthogonal to all topological dynamical systems of zero entropy.
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