Central limit theorems for generic lattice point counting

Abstract We consider the problem of counting lattice points contained in domains $$\mathbb {R}^d$$ R d defined by products linear forms. For $$d \ge 9$$ ≥ 9 we show that normalized discrepancies these problems satisfy non-degenerate Central Limit Theorems with respect to unique $${\text {SL}}_d(\mathbb {R})$$ SL ( ) -invariant probability measure on space unimodular lattices . also study more refined versions pertaining “spiraling approximations”. Our techniques are dynamical nature and exploit effective exponential mixing all orders for actions diagonalizable subgroups spaces lattices.