I am currently taking a number theory course. The professor who teaches the course is very well known in his research field which is Ergodic Theory. I find that he really likes to assign problems that deal with the density of a certain set. Sometimes the problems would be about proving whether the density of a certain set exists or not. Other times it would be proving whether the upper density of a given set is what he claims it is. Are there any general ways to tackling these kinds of problems?