• DOI:10.3934/dcds.2019108
• Corpus ID: 119678506

A generalization of Kátai's orthogonality criterion with applications

@article{Bergelson2019AGO,
  title={A generalization of K{\'a}tai's orthogonality criterion with applications},
  author={Vitaly Bergelson and Joanna Kułaga-Przymus and Mariusz Lema'nczyk and Florian Karl Richter},
  journal={Discrete \& Continuous Dynamical Systems - A},
  year={2019}
}

• V. Bergelson, J. Kułaga-Przymus, +1 author F. Richter
• Published 20 May 2017
• Mathematics
• Discrete & Continuous Dynamical Systems - A

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 Katai'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.