| Summary: | Abstract Baker, Harman, and Pintz showed that a weak form of the Prime Number Theorem holds in intervals of the form $$[x-x^{0.525},x]$$ [ 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 $$\delta \in [0.525,1]$$ δ ∈ [ 0.525 , 1 ] there exist positive integers k, d such that for sufficiently large x, the interval $$[x-x^\delta ,x]$$ [ x - x δ , x ] contains $$\gg _{k} \frac{x^\delta }{(\log x)^k}$$ ≫ k x δ ( 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.
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