In some deep sense, what you are saying is true : for example, if an infinite sequence of numbers is sufficiently dense, then it has arbitrarily long arithmetic

In some deep sense, what you are saying is true : for example, if an infinite sequence of numbers is sufficiently dense, then it has arbitrarily long arithmetic progressions.[1] On the other hand, Rankin in 1990 constructed a set with positive density which does not have long geometric progressions. [1] https://en.wikipedia.org/wiki/Szemer%C3%A9di%27s_theorem https://en.wikipedia.org/wiki/Szemer%C3%A9di%27s_theorem