Bounded gaps between primes in short intervals

TL;DR

This paper extends results on small gaps between primes to short intervals of length $x^eta$ for $eta ext{ in } [0.525,1]$, showing many prime pairs with bounded gaps exist in these intervals.

Contribution

It generalizes Maynard and Tao's small gaps results to short intervals, confirming a conjecture about prime gaps in intervals of length $x^eta$ for $eta ext{ in } [0.525,1].

Findings

Intervals of length $x^eta$ contain many prime pairs with bounded gaps.

Confirms Maynard's speculation on small gaps in short intervals.

Extends prime gap results to shorter intervals than previously known.

Abstract

Baker, Harman, and Pintz showed that a weak form of the Prime Number Theorem holds in intervals of the form $[x-x^{0.525},x]$ for large $x$. In this paper, we extend a result of Maynard and Tao concerning small gaps between primes to intervals of this length. More precisely, we prove that for any $\delta\in [0.525,1]$ there exist positive integers $k,d$ such that for sufficiently large $x$, the interval $[x-x^\delta,x]$ contains $\gg_{k} \frac{x^\delta}{(\log x)^k}$ pairs of consecutive primes differing by at most $d$. This confirms a speculation of Maynard that results on small gaps between primes can be refined to the setting of short intervals of this length.