The existence of small prime gaps in subsets of the integers

TL;DR

This paper extends the understanding of small prime gaps to various subsets of integers, showing that such gaps occur with positive density and are bounded, generalizing previous results beyond arithmetic progressions.

Contribution

It introduces new methods to prove the existence of small and bounded prime gaps in general sets of integers, broadening the scope of prior work focused on specific cases.

Findings

Small prime gaps occur with positive proportion in subsets of integers.

Bounded prime gaps are established for general classes of sets.

Results extend beyond arithmetic progressions to more general sets.

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