We will discuss multiplicative independence of sequences \(n\) and \(⌊nα⌋\) for irrational \(α\). In particular, we are going to show that for large class of arithmetic functions \(a, b : N → C\), the sequences \((a(n))n∈N\) and \((b(⌊nα⌋))n∈N\) are asymptotically uncorrelated. Furthermore, we will apply this result to show the logarithmic average of (λ(n)λ(⌊nα⌋))n∈N tends to 0 and that sequences (ω(n))n∈N and (ω(⌊nα⌋))n∈N behave as independent normally distributed random variables.

This is a joint work with Felipe Hernandez, Kevin Rizk, Khunpob Sereesuchart, and Ran Tao.