  We ask whether a stationary lattice in dimension $d$ whose points are shifted
by identically distributed but possibly dependent perturbations remains
hyperuniform. When $d = 1$ or $2$, we show that it is the case when the
perturbations have a finite $d$-moment, and that this condition is sharp. When
$d \geq 3$, we construct arbitrarily small perturbations such that the
resulting point process is not hyperuniform. As a side remark of independent
interest, we exhibit hyperuniform processes with arbitrarily slow decay of
their number variance.
