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Abstract Let$$\lambda $$ $\lambda$denote the Liouville function. We show that the logarithmic mean of$$\lambda (\lfloor \alpha _1n\rfloor )\lambda (\lfloor \alpha _2n\rfloor )$$ $\lambda \left(\lfloor {\alpha }_{1}n\rfloor \right)\lambda \left(\lfloor {\alpha }_{2}n\rfloor \right)$is 0 whenever$$\alpha _1,\alpha _2$$ ${\alpha }_1,{\alpha }_2$are positive reals with$$\alpha _1/\alpha _2$$ ${\alpha }_1/{\alpha }_2$irrational. We also show that for$$k\geqslant 3$$ $k⩾3$the logarithmic mean of$$\lambda (\lfloor \alpha _1n\rfloor )\cdots \lambda (\lfloor \alpha _kn\rfloor )$$ $\lambda \left(\lfloor {\alpha }_{1}n\rfloor \right)\cdots \lambda \left(\lfloor {\alpha }_{k}n\rfloor \right)$has some nontrivial amount of cancellation, under certain rational independence assumptions on the real numbers$$\alpha _i.$$ ${\alpha }_i.$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.