{
  "id": "math/0603415",
  "version": "v1",
  "published": "2006-03-16T23:18:14.000Z",
  "updated": "2006-03-16T23:18:14.000Z",
  "title": "On the determination of sets by their triple correlation in finite cyclic groups",
  "authors": [
    "Tamas Keleti",
    "Mihail N. Kolountzakis"
  ],
  "comment": "17 pages",
  "categories": [
    "math.CO",
    "math.NT"
  ],
  "abstract": "Let $G$ be a finite abelian group and $E$ a subset of it. Suppose that we know for all subsets $T$ of $G$ of size up to $k$ for how many $x \\in G$ the translate $x+T$ is contained in $E$. This information is collectively called the $k$-deck of $E$. One can naturally extend the domain of definition of the $k$-deck to include functions on $G$. Given the group $G$ when is the $k$-deck of a set in $G$ sufficient to determine the set up to translation? The 2-deck is not sufficient (even when we allow for reflection of the set, which does not change the 2-deck) and the first interesting case is $k=3$. We further restrict $G$ to be cyclic and determine the values of $n$ for which the 3-deck of a subset of $\\ZZ_n$ is sufficient to determine the set up to translation. This completes the work begun by Gr\\\"unbaum and Moore as far as the 3-deck is concerned. We additionally estimate from above the probability that for a random subset of $\\ZZ_n$ there exists another subset, not a translate of the first, with the same 3-deck. We give an exponentially small upper bound when the previously known one was $O(1\\bigl / \\sqrt{n})$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2006-03-16T23:18:14.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "43A25"
    ],
    "keywords": [
      "finite cyclic groups",
      "triple correlation",
      "determination",
      "finite abelian group",
      "exponentially small upper bound"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math......3415K"
    }
  }
}