On the multiplicative independence between $n$ and $\lfloor αn\rfloor$
In this article we investigate different forms of multiplicative independence between the sequences $n$ and $\lfloor n α\rfloor$ for irrational $α$. 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 $(b ( \lfloor αn \rfloor))_{n \in \mathbb{N}}$ are asymptotically uncorrelated. This new theorem is then applied to prove a $2$-dimensional version of the Erdős-Kac theorem, asserting that the sequences $(ω(n))_{n \in \mathbb{N}}$ and $(ω( \lfloor αn \rfloor)_{n\in \mathbb{N}}$ 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 $(λ(n) λ( \lfloor αn \rfloor))_{n \in \mathbb{N}}$ tends to $0$.