A generalization of K\'atai's orthogonality criterion with applications

Abstract

We study properties of arithmetic sets coming from multiplicative number theory and obtain applications in the theory of uniform distribution and ergodic theory. Our main theorem is a generalization of K\'atai's orthogonality criterion. Here is a special case of this theorem: Let $a\colon\mathbb{N}\to\mathbb{C}$ be a bounded sequence satisfying $$ \sum_{n\leq x} a(pn)\overline{a(qn)} = {\rm o}(x),~\text{for all distinct primes $p$ and $q$.} $$ Then for any multiplicative function $f$ and any $z\in\mathbb{C}$ the indicator function of the level set $E=\{n\in\mathbb{N}:f(n)=z\}$ satisfies $$ \sum_{n\leq x} \mathbb{1}_E(n)a(n)={\rm o}(x). $$ With the help of this theorem one can show that if $E=\{n_1 0$ with $c\notin \mathbb{N}$, $\log^r(t)$ for any $r>2$, $\log(\Gamma(t))$, $t\log(t)$, and $\frac{t}{\log t}$. The uniform distribution results, in turn, allow us to obtain new examples of ergodic sequences, i.e. sequences along which the ergodic theorem holds.