On a Bohr set analogue of Chowla's conjecture

TL;DR

This paper investigates the behavior of the Liouville function evaluated at Beatty sequences, establishing orthogonality and cancellation results under irrationality conditions, and extends these findings to general multiplicative functions.

Contribution

It generalizes Chowla's conjecture to Beatty sequences and provides new independence results for multiplicative functions evaluated at these sequences.

Findings

Logarithmic mean of Liouville function at two Beatty sequences is zero under irrational ratio.

Nontrivial cancellation occurs for products of Liouville functions at multiple Beatty sequences.

Bounds for logarithmic correlations of Liouville function along Bohr sets are established.

Abstract

Let $\lambda$ denote the Liouville function. We show that the logarithmic mean of $\lambda(\lfloor \alpha_1n\rfloor)\lambda(\lfloor \alpha_2n\rfloor)$ is $0$ whenever $\alpha_1,\alpha_2$ are positive reals with $\alpha_1/\alpha_2$ irrational. We also show that for $k\geq 3$ the logarithmic mean of $\lambda(\lfloor \alpha_1n\rfloor)\cdots \lambda(\lfloor \alpha_kn\rfloor)$ has some nontrivial amount of cancellation, under certain rational independence assumptions on the real numbers $\alpha_i$. Our results for the Liouville function generalise to produce independence statements for general bounded real-valued multiplicative functions evaluated at Beatty sequences. These results answer the two-point case of a conjecture of Frantzikinakis (and provide some progress on the higher order cases), generalising a recent result of Crn\v{c}evi\'c--Hern\'andez--Rizk--Sereesuchart--Tao. As an ingredient in our proofs, we establish bounds for the logarithmic correlations of the Liouville function along Bohr sets.