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A generalization of Kátai's orthogonality criterion with applications

  • * Corresponding author: Mariusz Lemańczyk

    * Corresponding author: Mariusz Lemańczyk 

The first author gratefully acknowledges the support of the NSF under grant DMS-1500575. The second author is supported by Narodowe Centrum Nauki UMO-2014/15/B/ST1/03736, the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No 647133 (ICHAOS)) and Foundation for Polish Science (FNP). The third author is supported by Narodowe Centrum Nauki UMO-2014/15/B/ST1/03736 and the EU grant "AOS", FP7-PEOPLE-2012-IRSES, No 318910

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  • 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 $ a\colon \mathbb{N} \to \mathbb{C} $ be a bounded sequence satisfying

    $ \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*} $

    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

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

    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 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.

    Mathematics Subject Classification: Primary: 11Nxx; Secondary: 37Axx, 11K06, 11J71, 11N25, 11N60.


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