{
  "id": "1904.07847",
  "version": "v1",
  "published": "2019-04-16T17:51:28.000Z",
  "updated": "2019-04-16T17:51:28.000Z",
  "title": "Distribution of determinant of sum of matrices",
  "authors": [
    "Daewoong Cheong",
    "Doowon Koh",
    "Thang Pham",
    "Anh Vinh Le"
  ],
  "categories": [
    "math.CO",
    "math.NT"
  ],
  "abstract": "Let $\\mathbb{F}_q$ be an arbitrary finite field of order $q$. In this article, we study $\\det S$ for certain types of subsets $S$ in the ring $M_2(\\mathbb F_q)$ of $2\\times 2$ matrices with entries in $\\mathbb F_q$. For $i\\in \\mathbb{F}_q$, let $D_i$ be the subset of $M_2(\\mathbb F_q)$ defined by $ D_i := \\{x\\in M_2(\\mathbb F_q): \\det(x)=i\\}.$ Then our results can be stated as follows. First of all, we show that when $E$ and $F$ are subsets of $D_i$ and $D_j$ for some $i, j\\in \\mathbb{F}_q^*$, respectively, we have $$\\det(E+F)=\\mathbb F_q,$$ whenever $|E||F|\\ge {15}^2q^4$, and then provide a concrete construction to show that our result is sharp. Next, as an application of the first result, we investigate a distribution of the determinants generated by the sum set $(E\\cap D_i) + (F\\cap D_j),$ when $E, F$ are subsets of the product type, i.e., $U_1\\times U_2\\subseteq \\mathbb F_q^2\\times \\mathbb F_q^2$ under the identification $ M_2(\\mathbb F_q)=\\mathbb F_q^2\\times \\mathbb F_q^2$. Lastly, as an extended version of the first result, we prove that if $E$ is a set in $D_i$ for $i\\ne 0$ and $k$ is large enough, then we have \\[\\det(2kE):=\\det(\\underbrace{E + \\dots + E}_{2k~terms})\\supseteq \\mathbb{F}_q^*,\\] whenever the size of $E$ is close to $q^{\\frac{3}{2}}$. Moreover, we show that, in general, the threshold $q^{\\frac{3}{2}}$ is best possible. Our main method is based on the discrete Fourier analysis.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-04-16T17:51:28.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "distribution",
      "determinant",
      "first result",
      "arbitrary finite field",
      "discrete fourier analysis"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}