  A discrete set in the $p$-dimensional Euclidian space is {\it almost
periodic}, if the measure with the unite masses at points of the set is almost
periodic in the weak sense. We propose to construct positive almost periodic
discrete sets as an almost periodic perturbation of a full rank discrete
lattice. Also we prove that each almost periodic discrete set on the real axes
is an almost periodic perturbation of some arithmetic progression.
  Next, we consider signed almost periodic discrete sets, i.e., when the signed
measure with masses $\pm1$ at points of a discrete set is almost periodic. We
construct a signed discrete set that is not almost periodic, while the
corresponding signed measure is almost periodic in the sense of distributions.
Also, we construct a signed almost periodic discrete set such that the measure
with masses +1 at all points of the set is not almost periodic.
