In this article 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 Erd\H{o}s-Kac theorem, asserting that the
sequences (ω(n))n∈N and (ω(⌊αn⌋)n∈N behave as independent normally distributed random
variables with mean loglogn and standard deviation loglogn.
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.Comment: 34 pages; fixed misspelled author name; December 7 2023: updated
authors affiliation, light edits, typos, added chart of main proof in
introductio