Close points on a modular hyperbola

TL;DR

This paper investigates the distribution of points on a modular hyperbola, establishing bounds on small squares containing multiple points and exploring special types of distances between these points.

Contribution

It provides a lower bound for the side length of small squares with multiple points on a modular hyperbola and analyzes the nature of distances between such points.

Findings

Lower bound for side length of squares with multiple points

Analysis of distances like primes, squarefree, and smooth numbers

Insights into point distribution on modular hyperbolas

Abstract

In this paper, we continue the study of small squares containing at least two points on a modular hyperbola $x y \equiv c \pmod{p}$. 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.