On the multiplicative independence between $n$ and $\lfloor \alpha n\rfloor$

TL;DR

This paper studies the multiplicative independence of sequences involving floors of irrational multiples, proving asymptotic uncorrelation and applying it to a 2D Erdős-Kac theorem and a variation of Chowla's conjecture.

Contribution

It establishes a general asymptotic uncorrelation result for sequences derived from irrational multiples and applies it to key problems in number theory.

Findings

Sequences are asymptotically uncorrelated for a large class of functions.

Proves a 2D Erdős-Kac theorem for the sequences involving floors of irrational multiples.

Shows a variation of Chowla's conjecture with logarithmic averages tending to zero.

Abstract

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 $0$.