The existence of small prime gaps in subsets of the integers

Abstract

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 $\left\{q_n|q_{n+1}-q_n\le \epsilon \log n\right\}$ 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.

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• Small gaps between products of two primes
• Bounded gaps between primes with a given primitive root
• Bounded Gaps Between Primes in Chebotarev Sets
• Bounded gaps between primes in number fields and function fields
• Primes in Tuples IV: Density of small gaps between consecutive primes