Uniform distribution of $\alpha \phi(n)$ and $\alpha \sigma(n)$ modulo 1, for non-Liouville numbers $\alpha$

There have been results on uniform distribution modulo 1 of sequences of the form $\{\alpha f(n)\}_{n=1}^\infty$, where $f(n)$ is an arithmetic function, and $\alpha$ is an irrational number. For example, $\{\alpha n \}_{n=1}^\infty$ (Bohl, Sierpiński and Weyl) and $\{\alpha \Omega(n)\}_{n=1}^\infty$ (Erdős and Delange) have been shown to be uniformly distributed modulo 1 for all irrational numbers $\alpha$. De Koninck and Kátai have shown that $\{\alpha \phi (n)\}_{n=1}^\infty$ and $\{\alpha \sigma (n)\}_{n=1}^\infty$ are uniformly distributed modulo 1 for a subset of irrational numbers $\alpha$. In this article, we will extend their result by showing that the sequences $\{\alpha \phi (n)\}_{n=1}^\infty$ and $\{\alpha \sigma (n)\}_{n=1}^\infty$ are uniformly distributed modulo 1 when $\alpha$ is a non-Liouville number. The proof will use Weyl's criterion, upper bounds of exponential functions established by Vinogradov and Vaughan, and the notion of a thin set established by Pollack and Vandehey. There are two corollaries that arise from the result of this article: $\{10^{\alpha \phi(n)}\}_{n=1}^\infty$ and $\{10^{\alpha \sigma(n)}\}_{n=1}^\infty$ are strong Benford sequences for all non-Liouville numbers $\alpha$, and the sequences $\{F(n) + \alpha \phi(n)\}_{n=1}^\infty$ and $\{F(n) + \alpha \sigma(n)\}_{n=1}^\infty$ are uniformly distributed modulo 1 for all non-Liouville numbers $\alpha$ and additive function $F$.