  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.
