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The existence of small prime gaps in subsets of the integers.


In this dissertation we consider the problem of finding small prime gaps in various sets $\mathcal{C} \subset \N$. To approach this problem we will consider collections of natural numbers that are well-controlled in arithmetic progressions. Following the work of Goldston-Pintz-Y{\i}ld{\i}r{\i}m, we let $q_n$ denote the $n$-th prime in $\mathcal{C}$ and establish that for some ``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 dissertation is to present results for general classes of sets.\

We also prove a second moment estimate for the Maynard-Tao sieve and give an application to Goldbach and de Polignac numbers. We show that at least one of two nice properties holds. Either consecutive Goldbach numbers lie within a finite distance from one another or else the set of de Polignac numbers has full density in $2 \N$.

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