  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.
