Weighted averages of arithmetic functions and applications to equidistribution and ergodic theory

For a wide range of functions $W\text{\hspace{0.17em}⁣}:\mathbb{N}\to \mathbb{N}$, we establish a general result for estimating weighted averages of the form$${\mathbb{E}}_{n\le N}^{W}f(\vartheta (n))=\frac{1}{W(N)}\sum _{n=1}^N(W(n)-W(n-1))f(\vartheta (n)),$$where $f\text{\hspace{0.17em}⁣}:\{1,\dots ,N\}\to \mathbb{C}$ is an arbitrary function, and $\vartheta (n)$ is any arithmetic function that adheres to a certain Gaussian distribution condition. (For instance, one may take $\vartheta (n)=\mathrm{\Omega }(n)$, where $\mathrm{\Omega }(n)$ counts the number of prime factors of $n$ with multiplicity, or $\vartheta (n)=s_q(p_n)$, where $s_q$ is the sum-of-digits function in base $q$ and $p_n$ denotes the $n$-th prime. Additional natural examples are discussed in the paper.) Building on our main theorem, we show that if $h(n)$ is a function from a Hardy field with polynomial growth then $(h(\vartheta (n)){)}_{n\in \mathbb{N}}$ is uniformly distributed mod $1$ if and only if one of the following (mutually exclusive) conditions is satisfied: (i) ${\lim }_{x\to \mathrm{\infty }}\frac{\mathrm{\mid }h(x)-p(x)\mathrm{\mid }}{x\log x}=\mathrm{\infty }$ for all $p(x)\in \mathbb{Q}[x]$; (ii) ${\lim }_{x\to \mathrm{\infty }}\frac{\mathrm{\mid }h(x)-p(x)\mathrm{\mid }}{\sqrt{x}}=\mathrm{\infty }$ for each $p(x)\in \mathbb{Q}[x]$ and there exists $q(x)\in \mathbb{Q}[x]$ such that ${\lim }_{x\to \mathrm{\infty }}\frac{\mathrm{\mid }h(x)-q(x)\mathrm{\mid }}{x}<\mathrm{\infty }$. This leads to several novel applications. For example, it follows that $(\mathrm{\Omega }(n{)}^c{)}_{n\in \mathbb{N}}$ is uniformly distributed mod $1$ if and only if $c$ is a non-integer greater than $\frac{1}{2}$.