Counting lattice points and weak admissibility of a lattice and its dual

Abstract

We prove a counting theorem concerning the number of lattice points for the dual lattices of weakly admissible lattices in an inhomogeneously expanding box. The error term is expressed in terms of a certain function ν(Γ⊥, ·) of the dual lattice Γ⊥, and we carefully analyse the relation of this quantity with ν(Γ, ·). In particular, we show that ν(Γ⊥, ·) = ν(Γ, ·) for any unimodular lattice of rank 2, but that for higher ranks it is in general not possible to bound one function in terms of the other. This result relies on Beresnevich’s recent breakthrough on Davenport’s problem regarding badly approximable points on submanifolds of ℝn. Finally, we apply our counting theorem to establish asymptotics for the number of Diophantine approximations with bounded denominator as the denominator bound gets large.