Limit points and long gaps between primes

TL;DR

This paper demonstrates that the set of limit points of normalized prime gaps contains at least 25% of all nonnegative real numbers, using advanced results on prime gaps and their distributions.

Contribution

It establishes a significant lower bound on the density of limit points of normalized prime gaps, extending previous work with new probabilistic and analytical techniques.

Findings

At least 25% of nonnegative real numbers are limit points of normalized prime gaps.

The result holds for functions growing more slowly than the Erdős–Rankin function.

The proof combines recent breakthroughs on bounded and long prime gaps.

Abstract

Let $d_n = p_{n+1} - p_n$, where $p_n$ denotes the $n$th smallest prime, and let $R(T) = \log T \log_2 T\log_4 T/(\log_3 T)^2$ (the "Erd{\H o}s--Rankin" function). We consider the sequence $(d_n/R(p_n))$ of normalized prime gaps, and show that its limit point set contains at least $25\%$ of nonnegative real numbers. We also show that the same result holds if $R(T)$ is replaced by any "reasonable" function that tends to infinity more slowly than $R(T)\log_3 T$. We also consider "chains" of normalized prime gaps. Our proof combines breakthrough work of Maynard and Tao on bounded gaps between primes with subsequent developments of Ford, Green, Konyagin, Maynard and Tao on long gaps between consecutive primes.