Central limit theorems for generic lattice point counting

Abstract

We consider the problem of counting lattice points contained in domains in mathbb {R}^d defined by products of linear forms. For d ge 9 we show that the normalized discrepancies in these counting problems satisfy non-degenerate Central Limit Theorems with respect to the unique {text {SL}}_d(mathbb {R})-invariant probability measure on the space of unimodular lattices in mathbb {R}^d. 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 diagonalizable subgroups on spaces of unimodular lattices.