We consider the problem of finding small prime gaps in various sets of integers $\mathcal{ C}$. Following the work of Goldston-Pintz-Yildirim, we will consider collections of natural numbers that are well-controlled in arithmetic progressions. Letting $q_n$ denote the $n$-th prime in $\mathcal{ C}$, we will establish that for any small constant $\epsilon>0$, the set $\left\{ q_n| q_{ n+1}-q_n \leq \epsilon \log n \right\}$ constitutes a positive proportion of all prime numbers. Using the techniques developed by Maynard and Tao we will also demonstrate that $\mathcal{ C}$ has bounded prime gaps. Specific examples, such as the case where $\mathcal{ C}$ is an arithmetic progression have already been studied and so the purpose of this paper is to present results for general classes of sets.
Categories: math.NT