Limit points of normalized prime gaps

Abstract

We show that at least 1/3 of positive real numbers are in the set of limit points of normalized prime gaps. More precisely, if $p_n$ denotes the $n$th prime and $\mathbb{L}$ is the set of limit points of the sequence $\{(p_{n+1}-p_n)/\log p_n\}_{n=1}^\infty,$ then for all $T\geq 0$ the Lebesque measure of $\mathbb{L} \cap [0,T]$ is at least $T/3.$ This improves the result of Pintz (2015) that the Lebesque measure of $\mathbb{L} \cap [0,T]$ is at least $(1/4-o(1))T,$ which was obtained by a refinement of the previous ideas of Banks, Freiberg, and Maynard (2015). Our improvement comes from using Chen's sieve to give, for a certain sum over prime pairs, a better upper bound than what can be obtained using Selberg's sieve. Even though this improvement is small, a modification of the arguments Pintz and Banks, Freiberg, and Maynard shows that this is sufficient. In addition, we show that there exists a constant $C$ such that for all $T \geq 0$ we have $\mathbb{L} \cap [T,T+C] \neq \emptyset,$ that is, gaps between limit points are bounded by an absolute constant.