On the multiplicative independence between $n$ and $\lfloor \alpha n\rfloor$

Abstract

In this article we investigate different forms of multiplicative independence between the sequences $n$ and $\lfloor n\alpha \rfloor$ for irrational $\alpha$. Our main theorem shows that for a large class of arithmetic functions $a,b:\mathbb{N}\to \mathbb{C}$ the sequences $(a(n){)}_{n\in \mathbb{N}}$ and $(b(\lfloor \alpha n\rfloor ){)}_{n\in \mathbb{N}}$ are asymptotically uncorrelated. This new theorem is then applied to prove a $2$-dimensional version of the Erdős-Kac theorem, asserting that the sequences $(\omega (n){)}_{n\in \mathbb{N}}$ and $(\omega (\lfloor \alpha n\rfloor {)}_{n\in \mathbb{N}}$ behave as independent normally distributed random variables with mean $\log \log n$ and standard deviation $\sqrt{\log \log n}$. Our main result also implies a variation on Chowla's Conjecture asserting that the logarithmic average of $(\lambda (n)\lambda (\lfloor \alpha n\rfloor ){)}_{n\in \mathbb{N}}$ tends to $0$.

34 pages; fixed misspelled author name; December 7 2023: updated authors affiliation, light edits, typos, added chart of main proof in introduction