标题： 关于斯克里加诺夫的计数定理 显示英文标题

标题： On a Counting Theorem of Skriganov

摘要： 我们证明了一个关于非均匀扩张方箱中弱可接受格的对偶格的格点数的计数定理，该定理推广了Skriganov的一个计数定理。 误差项用对偶格$\Gamma^\perp$的某个函数$\nu(\Gamma^\perp,\cdot)$表示，我们仔细分析了该量与$\nu(\Gamma,\cdot)$的关系。 特别是，我们证明了对于任意秩为2的单模格，$\nu(\Gamma^\perp,\cdot)=\nu(\Gamma,\cdot)$成立，但对于更高秩的情况，通常无法用一个函数来界定另一个函数。 最后，我们将我们的计数定理应用于建立分母有界时的丢番图逼近数量的渐进行为，当分母界限变大时。 显示英文摘要

摘要： 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.