On the multiplicative independence between n and ⌊αn⌋

We investigate different forms of multiplicative independence between the sequences n and ⌊nα⌋ for irrational α. Our main theorem shows that for a large class of arithmetic functions a, b: N → C the sequences (a(n))n∈N and (b(⌊αn⌋))n∈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∈N and (ω(⌊αn⌋))n∈N behave as independent normally distributed random variables with mean log log n and standard deviation √log log n. Our main result also implies a variation on Chowla’s conjecture asserting that the logarithmic average of (λ(n)λ(⌊αn⌋))n∈N tends to 0.