  We consider the problem of counting lattice points contained in domains in
$\mathbb{R}^d$ defined by products of linear forms and we show that the
normalized discrepancies in these counting problems satisfy non-degenerate
Central Limit Theorems, provided that $d \geq 9$.
  We also study more refined versions pertaining to "spiraling of
approximations". Our techniques are dynamical in nature and exploit effective
exponential mixing of all orders for actions of higher-rank abelian groups on
the space of unimodular lattices.
