  In any dimension $d \geq 2$, there is no known example of a low-discrepancy
sequence which possess Poisssonian pair correlations. This is in some sense
rather surprising, because low-discrepancy sequences always have
$\beta$-Poissonian pair correlations for all $0 < \beta < \tfrac{1}{d}$ and are
therefore arbitrarily close to having Poissonian pair correlations (which
corresponds to the case $\beta = \tfrac{1}{d}$). In this paper, we further
elaborate on the closeness of the two notions. We show that $d$-dimensional
Kronecker sequences for badly approximable vectors $\vec{\alpha}$ with an
arbitrary small uniformly distributed stochastic error term generically have
$\beta = \tfrac{1}{d}$-Poissonian pair correlations.
