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.

中文翻译：

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计算格点和格及其对偶的弱可容许性

我们证明了一个关于非均匀扩展盒中弱可容许格的对偶格点数的计数定理。误差项用对偶点阵Γ⊥的某个函数ν(Γ⊥,·)表示，我们仔细分析了这个量与ν(Γ,·)的关系。特别是，我们证明 ν(Γ⊥, ·) = ν(Γ, ·) 对于任何 2 阶的单模晶格，但对于更高的阶，通常不可能将一个函数绑定到另一个函数。这个结果依赖于 Beresnevich 最近在 Davenport 问题上的突破，该问题涉及 ℝn 的子流形上的极近似点。最后，当分母边界变大时，我们应用我们的计数定理来建立具有有界分母的丢番图近似的数量的渐近性。