In Diophantine geometry, determining the abundance of integral and rational points is a crucial topic. We have studied "$S$-integral points," which are intermediate between integral and rational points, and have proven that certain Zariski open sets in $\mathbb{P}^n$ are abundant in $S$-integral points in the sense of "potential density". In this talk, we will explain this result with an introduction to the theory of potential density of $S$-integral points.