{
  "id": "1707.05437",
  "version": "v1",
  "published": "2017-07-18T02:00:47.000Z",
  "updated": "2017-07-18T02:00:47.000Z",
  "title": "Bounded gaps between primes in short intervals",
  "authors": [
    "Ryan Alweiss",
    "Sammy Luo"
  ],
  "comment": "29 pages",
  "categories": [
    "math.NT"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-07-18T02:00:47.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11N05",
      "11N36"
    ],
    "keywords": [
      "short intervals",
      "bounded gaps",
      "prime number theorem holds",
      "tao concerning small gaps",
      "weak form"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 29,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}