We give a precise estimate for the number of lattice points in certain bounded subsets of $\mathbb{R}^{n}$ that involve `hyperbolic spikes' and occur naturally in multiplicative Diophantine approximation. We use Wilkie's o-minimal structure $\mathbb{R}_{\exp}$ and expansions thereof to formulate our counting result in a general setting. We give two different applications of our counting result. The first one establishes nearly sharp upper bounds for sums of reciprocals of fractional parts, and thereby sheds light on a question raised by Le and Vaaler, extending previous work of Widmer and of the author. The second application establishes new examples of linear subspaces of Khintchine type thereby refining a theorem by Huang and Liu. For the proof of our counting result we develop a sophisticated partition method which is crucial for further upcoming work on sums of reciprocals of fractional parts over distorted boxes.

中文翻译：

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关于弱可容许格的计数定理

我们对 $\mathbb{R}^{n}$ 的某些有界子集中的格点数量进行了精确估计，这些子集涉及“双曲线尖峰”并在乘法丢番图近似中自然出现。我们使用 Wilkie 的 o 极小结构 $\mathbb{R}_{\exp}$ 及其扩展来制定我们在一般设置中的计数结果。我们给出了计数结果的两种不同应用。第一个为分数部分的倒数之和建立了近乎尖锐的上限，从而阐明了 Le 和 Vaaler 提出的问题，扩展了 Widmer 和作者之前的工作。第二个应用程序建立了 Khintchine 类型的线性子空间的新例子，从而改进了 Huang 和 Liu 的定理。