{
  "id": "1902.04176",
  "version": "v1",
  "published": "2019-02-11T23:05:50.000Z",
  "updated": "2019-02-11T23:05:50.000Z",
  "title": "Bivariate fluctuations for the number of arithmetic progressions in random sets",
  "authors": [
    "Yacine Barhoumi-Andréani",
    "Christoph Koch",
    "Hong Liu"
  ],
  "comment": "30 pages",
  "categories": [
    "math.PR",
    "math.CO"
  ],
  "abstract": "We study arithmetic progressions $\\{a,a+b,a+2b,\\dots,a+(\\ell-1) b\\}$, with $\\ell\\ge 3$, in random subsets of the initial segment of natural numbers $[n]:=\\{1,2,\\dots, n\\}$. Given $p\\in[0,1]$ we denote by $[n]_p$ the random subset of $[n]$ which includes every number with probability $p$, independently of one another. The focus lies on sparse random subsets, i.e.\\ when $p=p(n)=o(1)$ as $n\\to+\\infty$. Let $X_\\ell$ denote the number of distinct arithmetic progressions of length $\\ell$ which are contained in $[n]_p$. We determine the limiting distribution for $X_\\ell$ not only for fixed $\\ell\\ge 3$ but also when $\\ell=\\ell(n)\\to+\\infty$. The main result concerns the joint distribution of the pair $(X_{\\ell},X_{\\ell'})$, $\\ell>\\ell'$, for which we prove a bivariate central limit theorem for a wide range of $p$. Interestingly, the question of whether the limiting distribution is trivial, degenerate, or non-trivial is characterised by the asymptotic behaviour (as $n\\to+\\infty$) of the threshold function $\\psi_\\ell=\\psi_\\ell(n):=np^{\\ell-1}\\ell$. The proofs are based on the method of moments and combinatorial arguments, such as an algorithmic enumeration of collections of arithmetic progressions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-02-11T23:05:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60C05",
      "11B25",
      "05C80",
      "60F05"
    ],
    "keywords": [
      "random sets",
      "bivariate fluctuations",
      "bivariate central limit theorem",
      "distinct arithmetic progressions",
      "study arithmetic progressions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}