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 Kátai's orthogonality criterion. Here is a special case of this theorem:

Theorem. Let \begin{document}$ a\colon \mathbb{N} \to \mathbb{C} $\end{document} be a bounded sequence satisfying

\begin{document}$ \begin{equation*} \sum\limits_{n\leq x} a(pn)\overline{a(qn)} = {\rm{o}} (x),\;\; {\mathit{for\;all\;distinct\;primes}\;\; p \;\;\mathit{and}\;\; q}. \end{equation*} $\end{document}

Then for any multiplicative function \begin{document}$ f $\end{document} and any \begin{document}$ z\in \mathbb{C} $\end{document} the indicator function of the level set \begin{document}$ E = \{n\in \mathbb{N} :f(n) = z\} $\end{document} satisfies

\begin{document}$ \begin{equation*} \sum\limits_{n\leq x} {1_{E}(n)a(n)} = {\rm{o}} (x). \end{equation*} $\end{document}

With the help of this theorem one can show that if \begin{document}$ E = \{n_1<n_2<\ldots\} $\end{document} is a level set of a multiplicative function having positive 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.