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 $T⩾0$ the Lebesque measure of $\mathbb{L}\cap [0,T]$ is at least $T/3$. This improves the result of Pintz from 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 from 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 given by 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 $T⩾0$ we have $\mathbb{L}\cap [T,T+C]\ne \varnothing$, that is, gaps between limit points are bounded by an absolute constant.

中文翻译：

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标准化素数间隙的极限点

我们表明，至少有1/3的正实数在归一化质数间隙的极限点集中。更确切地说，如果$p_{ñ}$ 表示 $ñ$素数和 $\mathbb{大号}$ 是序列的极限点的集合 ${\{（p_{ñ+\mathrm{1个}}-p_{ñ}）/\mathrm{日志}{p}_{ñ}\}}_{ñ=\mathrm{1个}}^{\infty }$，那么对于所有人 $Ť⩾0$ 勒贝克量度 $\mathbb{大号}\cap [0，Ť]$ 至少是 $Ť/3$。Lebesque测算的2015年Pintz的结果得到了改善$\mathbb{大号}\cap [0，Ť]$ 至少是 $（\mathrm{1个}/4-Ø（\mathrm{1个}））Ť$，是从2015年对Banks，Freiberg和Maynard的先前想法进行的改进而获得的。我们的改进来自于使用Chen的筛子，对于一定数量的质数对，其上限比使用Selberg's的上限更好。筛。即使这种改进很小，但是对Pintz和Banks，Freiberg和Maynard给出的论点进行的修改表明，这已经足够。另外，我们表明存在一个常数$C$ 这样对于所有人 $Ť⩾0$ 我们有 $\mathbb{大号}\cap [Ť，Ť+C]\ne \varnothing$即，极限点之间的间隙以绝对常数为界。