Close points on a modular hyperbola

In this paper, we continue the study of small squares containing at least two points on a modular hyperbola $xy\equiv c(\mathrm{m}\mathrm{o}\mathrm{d}p)$. We deduce a lower bound for its side length. We also investigate what happens if the ``distances" between two such points are special type of numbers like prime numbers, squarefree numbers or smooth numbers as well as more general multiplicatively closed sets or almost dense sets.