Bounded gaps between primes in short intervals

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 $\delta\in [0.525,1]$thereexistpositiveintegers$k,d$suchthatforsufficientlylarge$x$,theinterval$[x-x^\delta,x]$contains$\gg_{k} \frac{x^\delta}{(\log x)^k}$pairsofconsecutiveprimesdifferingbyatmost$d$. 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.