A generalization of Kátai'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 $E={n_1<n_2<\ldots}$ 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.