On a Bohr set analogue of Chowla's conjecture

Abstract

Let $λ$ denote the Liouville function. We show that the logarithmic mean of $λ(\lfloor α_1n\rfloor)λ(\lfloor α_2n\rfloor)$ is $0$ whenever $α_1,α_2$ are positive reals with $α_1/α_2$ irrational. We also show that for $k\geq 3$ the logarithmic mean of $λ(\lfloor α_1n\rfloor)\cdots λ(\lfloor α_kn\rfloor)$ has some nontrivial amount of cancellation, under certain rational independence assumptions on the real numbers $α_i$. 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.