Bergelson, VitalyKułaga-Przymus, JoannaLemańczyk, MariuszRichter, Florian Karl2021-11-262021-11-262021-11-26201910.3934/dcds.2019108https://infoscience.epfl.ch/handle/20.500.14299/1832461705.07322<p style='text-indent:20px;'>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:</p><p style='text-indent:20px;'><b>Theorem.</b> <i>Let</i> <inline-formula><tex-math id="M2">\begin{document}$ a\colon \mathbb{N} \to \mathbb{C} $\end{document}</tex-math></inline-formula> <i>be a bounded sequence satisfying</i></p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \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*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'><i>Then for any multiplicative function</i> <inline-formula><tex-math id="M3">\begin{document}$ f $\end{document}</tex-math></inline-formula> <i>and any</i> <inline-formula><tex-math id="M4">\begin{document}$ z\in \mathbb{C} $\end{document}</tex-math></inline-formula> <i>the indicator function of the level set</i> <inline-formula><tex-math id="M5">\begin{document}$ E = \{n\in \mathbb{N} :f(n) = z\} $\end{document}</tex-math></inline-formula> <i>satisfies</i></p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ \begin{equation*} \sum\limits_{n\leq x} {1_{E}(n)a(n)} = {\rm{o}} (x). \end{equation*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>With the help of this theorem one can show that if <inline-formula><tex-math id="M6">\begin{document}$ E = \{n_1&lt;n_2&lt;\ldots\} $\end{document}</tex-math></inline-formula> 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 $c &gt; 0$ with $c\notin \mathbb{N} $, $\log^r(t)$ for any $r &gt; 2$, $\log(\Gamma(t))$, $t\log(t)$, 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.</p>text::journal::journal article::research article