{
  "id": "1902.00353",
  "version": "v1",
  "published": "2019-02-01T14:26:29.000Z",
  "updated": "2019-02-01T14:26:29.000Z",
  "title": "A counterexample to a strong variant of the Polynomial Freiman-Ruzsa conjecture",
  "authors": [
    "James Aaronson"
  ],
  "comment": "3 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $p$ be a prime. One formulation of the Polynomial Freiman-Ruzsa conjecture over $\\mathbb{F}_p$ can be stated as follows. If $\\phi : \\mathbb{F}_p^n \\rightarrow \\mathbb{F}_p^N$ is a function such that $\\phi(x+y) - \\phi(x) - \\phi(y)$ takes values in some set $S$, then there is a linear map $\\tilde{\\phi} : \\mathbb{F}_p^n \\rightarrow \\mathbb{F}_p^N$ with the property that $\\phi - \\tilde{\\phi}$ takes at most $|S|^{O(1)}$ values. A strong variant of this conjecture states that, in fact, there is a linear map $\\tilde{\\phi}$ such that $\\phi - \\tilde{\\phi}$ takes values in $tS$ for some constant $t$. In this note, we discuss a counterexample to this conjecture.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-02-01T14:26:29.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "polynomial freiman-ruzsa conjecture",
      "strong variant",
      "counterexample",
      "linear map",
      "conjecture states"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 3,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}