  We generalise M. M. Skriganov's notion of weak admissibility for lattices to
include standard lattices occurring in Diophantine approximation and algebraic
number theory, and we prove estimates for the number of lattice points in sets
such as aligned boxes. Our result improves on Skriganov's celebrated counting
result if the box is sufficiently distorted, the lattice is not admissible,
and, e.g., symplectic or orthogonal. We establish a criterion under which our
error term is sharp, and we provide examples in dimensions $2$ and $3$ using
continued fractions. We also establish a similar counting result for primitive
lattice points, and apply the latter to the classical problem of Diophantine
approximation with primitive points as studied by Chalk, Erd\H{o}s, and others.
Finally, we use o-minimality to describe large classes of sets
