On a Counting Theorem of Skriganov

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, which generalises a counting theorem of Skriganov. The error term is expressed in terms of a certain function \(\nu(\Gamma^\perp,\cdot)\) of the dual lattice \(\Gamma^\perp\), and we carefully analyse the relation of this quantity with \(\nu(\Gamma,\cdot)\). In particular, we show that \(\nu(\Gamma^\perp,\cdot)=\nu(\Gamma,\cdot)\) 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. Finally, we apply our counting theorem to establish asymptotics for the number of Diophantine approximations with bounded denominator as the denominator bound gets large.