Analytic Number Theory has witnessed many breakthroughs on fundamental problems in recent years, including Zhang, Maynard and Tao’s celebrated results on bounded prime gaps. In this talk, I will explain that given m > 1 and a sufficiently large q, how to adapt their method to establish the existence of an arithmetic progression with common difference q for which the m-th least prime in such progression is ≪m q, a result that can be seen to be best possible. As we vary over progressions instead of fixing a particular one, the nature of our problem differs from others in the literature. Furthermore, inspired by the simple yet far-reaching Poincar ́e recurrence theorem, I will discuss a generalization to measure-preserving systems. This is an on-going joint work with Tony Haddad and Cihan Sabuncu.