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

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\colon \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 $0$.