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- x0.525, 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 δ∈ [ 0.525 , 1 ] there exist positive integers k, d such that for sufficiently large x, the interval [ x- xδ, x] contains ≫kxδ(logx)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. © 2018, SpringerNature.