# On a counting theorem for weakly admissible lattices

## TL;DR

This paper provides a precise lattice point counting estimate in regions with hyperbolic spikes using o-minimal structures, with applications to Diophantine approximation and Khintchine type subspaces.

## Contribution

It introduces a general counting theorem for weakly admissible lattices in regions with hyperbolic spikes using o-minimal structures, and applies it to Diophantine approximation and Khintchine type subspaces.

## Key findings

- Established nearly sharp bounds for sums of reciprocals of fractional parts.
- Extended previous results by Lê, Vaaler, Widmer, Huang, and Liu.
- Developed a partition method for counting lattice points in complex regions.

## Abstract

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 L\^e 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.