One way of looking at things like this is from the perspective of so called Ramsey theory. Notions like syndenticity(the property of a set having bounded gaps), upper density(the lim-sup of the proportion of numbers in the set), thickness(the property of having arbitrarily large successive subsequences), etc. all capture different notions of this.
Edit: let’s not forget the rich theory involved in examining if sets/sequences contain arithmetic (or geometric) progressions! The well known Green-Tao theorem about primes containing arbitrarily long arithmetic progressions is one such result. In fact, the real Green-Tao theorem says that any set which has nonzero upper density with respect to the set of primes contains arbitrarily long arithmetic progressions.
Edit: let’s not forget the rich theory involved in examining if sets/sequences contain arithmetic (or geometric) progressions! The well known Green-Tao theorem about primes containing arbitrarily long arithmetic progressions is one such result. In fact, the real Green-Tao theorem says that any set which has nonzero upper density with respect to the set of primes contains arbitrarily long arithmetic progressions.