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

Abstract

For a wide range of functions $W:\mathbb{N}\to \mathbb{N}$, we establish a general result for estimating weighted averages of the form\[\mathbb{E}^{W}_{n \le N} f(\vartheta(n))= \frac{1}{W(N)}\sum_{n=1}^N (W(n)-W(n-1))f(\vartheta(n)),\]where $f:\{1,\dots ,N\}\to \mathbb{C}$ is an arbitrary function, and $\vartheta (n)$ is any arithmetic function that adheres to a certain Gaussian distribution condition. (For instance, one may take $\vartheta (n)=\Omega (n)$, where $\Omega (n)$ 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) ${\lim }_{x\to \infty }\frac{|h(x)-p(x)|}{x\log x}=\infty$ for all $p(x)\in \mathbb{Q}[x]$; (ii) ${\lim }_{x\to \infty }\frac{|h(x)-p(x)|}{\sqrt{x}}=\infty$ for each $p(x)\in \mathbb{Q}[x]$ and there exists $q(x)\in \mathbb{Q}[x]$ such that ${\lim }_{x\to \infty }\frac{|h(x)-q(x)|}{x}<\infty$. This leads to several novel applications. For example, it follows that $(\Omega (n{)}^c{)}_{n\in \mathbb{N}}$ is uniformly distributed mod $1$ if and only if $c$ is a non-integer greater than $\frac{1}{2}$.

arXiv comment32 pages, Updated version: Theorem 1.7 added as another application