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The existence of small prime gaps in subsets of the integers.
 Benatar, Jacques
 Advisor(s): Tao, Terence C
Abstract
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 wellcontrolled in arithmetic progressions. Following the work of GoldstonPintzY{\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 MaynardTao 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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