Title: Bounded gaps between primes in short intervals,
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.