{
  "id": "1004.3709",
  "version": "v1",
  "published": "2010-04-21T13:53:15.000Z",
  "updated": "2010-04-21T13:53:15.000Z",
  "title": "Freiman homomorphisms of random subsets of $\\mathbb{Z}_{N}$",
  "authors": [
    "Gonzalo Fiz Pontiveros"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $A$ be a random subset of $\\mathbb{Z}_{N}$ obtained by including each element of $\\mathbb{Z}_{N}$ in $A$ independently with probability $p$. We say that $A$ is \\emph{linear} if the only Freiman homomorphisms are given by the restrictions of functions of the form $f(x)= ax+b$. For which values of $p$ do we have that $A$ is linear with high probability as $N\\to\\infty$ ? First, we establish a geometric characterisation of linear subsets. Second, we show that if $p=o(N^{-2/3})$ then $A$ is not linear with high probability whereas if $p=N^{-1/2+\\epsilon}$ for any $\\epsilon>0$ then $A$ is linear with high probability.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-04-21T13:53:15.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "freiman homomorphisms",
      "random subset",
      "high probability",
      "geometric characterisation",
      "linear subsets"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1004.3709F"
    }
  }
}