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

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 $E=\{n_1<n_2<\ldots\}$ is a level set of a multiplicative function having positive upper density, then for a large class of sufficiently smooth functions $h\colon(0,\infty)\to\mathbb{R}$ the sequence $(h(n_j))_{j\in\mathbb{N}}$ is uniformly distributed $\bmod~1$. This class of functions $h(t)$ includes: all polynomials $p(t)=a_kt^k+\ldots+a_1t+a_0$ such that at least one of the coefficients $a_1,a_2,\ldots,a_k$ is irrational, $t^c$ for any $c>0$ 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.