  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
$a, b \colon \mathbb{N} \to \mathbb{C}$ 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 Erd\H{o}s-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 $0$.
