Uniform distribution of polynomially-defined additive functions to varying moduli

There have been results on uniform distribution modulo 1 of sequences of the form [Formula: see text] where [Formula: see text] is an arithmetic function and [Formula: see text] is an irrational number. For example, [Formula: see text] (Bohl, Sierpiński and Weyl) and [Formula: see text] (Erdős and Delange) have been shown to be uniformly distributed modulo 1 for all irrational numbers [Formula: see text]. De Koninck and Kátai have shown that [Formula: see text] and [Formula: see text] are uniformly distributed modulo 1 for a subset of irrational numbers [Formula: see text]. In this article, we will extend their result by showing that the sequences [Formula: see text] and [Formula: see text] are uniformly distributed modulo 1 when [Formula: see text] 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: [Formula: see text] and [Formula: see text] are strong Benford sequences for all non-Liouville numbers [Formula: see text], and the sequences [Formula: see text] and [Formula: see text] are uniformly distributed modulo 1 for all non-Liouville numbers [Formula: see text] and additive function F.