On a Bohr set analogue of Chowla's conjecture

Abstract

Let $\lambda$ denote the Liouville function. We show that the logarithmic mean of $\lambda (\lfloor {\alpha }_{1}n\rfloor )\lambda (\lfloor {\alpha }_{2}n\rfloor )$ is $0$ whenever ${\alpha }_1,{\alpha }_2$ are positive reals with ${\alpha }_1/{\alpha }_2$ irrational. We also show that for $k\ge 3$ the logarithmic mean of $\lambda (\lfloor {\alpha }_{1}n\rfloor )\cdots \lambda (\lfloor {\alpha }_{k}n\rfloor )$ 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 Crnčević--Hernández--Rizk--Sereesuchart--Tao. As an ingredient in our proofs, we establish bounds for the logarithmic correlations of the Liouville function along Bohr sets.