Counting in generic lattices and higher rank actions

We consider the problem of counting lattice points contained in domains in R 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 ≥ 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.

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