{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $\\F_q$ be a [[finite field|0ZXZ82FTX2]] with $q$ elements and let $n \\ge 1$.\nA *Kakeya set* in $\\F_q^n$ is a set $K \\subseteq \\F_q^n$ such that for every $x \\in \\F_q^n \\setminus \\set{0}$ there is $y \\in \\F_q^n$ with $\\set{y + t x : t \\in \\F_q} \\subseteq K$.\nThen every Kakeya set $K$ in $\\F_q^n$ satisfies\n$$\\abs{K} \\ge \\binom{q + n - 1}{n} \\ge \\frac{q^n}{n!}.$$\n\n## Remarks\n\nThis is Dvir's theorem, which answered the finite field analogue of the [[Kakeya problem|6PEN7AKC7M]] asked by Thomas Wolff.\nThe proof uses the polynomial method.\nIf $\\abs{K} \u003c \\binom{q + n - 1}{n}$, the number of monomials of degree at most $q - 1$ in $n$ variables, then some nonzero [[polynomial|07GNX8GCYP]] $P$ of degree $d \\le q - 1$ vanishes on $K$.\nLet $P_d$ be the top-degree homogeneous part of $P$.\nRestricting $P$ to a line $y + t x$ inside $K$ gives a polynomial in $t$ of degree at most $d \u003c q$ with $q$ roots, so it is zero.\nIts coefficient of $t^d$ is $P_d(x)$, so $P_d(x) = 0$.\nSo $P_d$ vanishes on all of $\\F_q^n$, which is impossible for a nonzero polynomial of degree less than $q$ in each variable.\n\nFor $n = 1$ the bound gives $\\abs{K} \\ge q$, and indeed $K$ must be the whole line $\\F_q$.\nFor $n = 2$ it gives $\\abs{K} \\ge q(q + 1)/2$.\n\n## Explanations\n\n### L1 human\n\nA Kakeya set is a set that contains a full line pointing in every direction.\nOver the real numbers such sets can have measure zero, which is surprising.\nOver a finite field the theorem says the opposite: such a set must contain a fixed positive fraction, about $1/n!$, of all the points of the space.\n\n### L2 human\n\nDvir's proof is a few lines long and made the polynomial method a standard tool in combinatorics.\nThe idea is that a set too small cannot avoid being the zero set of a low-degree polynomial, while lines in all directions force the polynomial to vanish identically.\nThe constant $1/n!$ has since been improved, but the order $q^n$ is optimal.\nThe finite field result does not solve the Euclidean Kakeya conjecture, whose difficulty comes from the multiscale structure of $\\R^n$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "about": [
      "6PEN7AKC7M"
    ],
    "kind": "statement",
    "msc": [
      "51E20",
      "05B25",
      "11T06"
    ],
    "name": "Kakeya problem over finite fields",
    "role": "theorem",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Kakeya_set"
      }
    ],
    "tag": "KSW18V47ZD",
    "uses": [
      "0ZXZ82FTX2",
      "07GNX8GCYP",
      "6PEN7AKC7M"
    ]
  },
  "kind": "statement",
  "link": "https://tamnd.github.io/math2/node/KSW18V47ZD/",
  "links": [
    {
      "field": "about",
      "target": "6PEN7AKC7M"
    },
    {
      "field": "uses",
      "target": "0ZXZ82FTX2"
    },
    {
      "field": "uses",
      "target": "07GNX8GCYP"
    },
    {
      "field": "uses",
      "target": "6PEN7AKC7M"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/KSW18V47ZD.md",
  "msc": [
    "51E20",
    "05B25",
    "11T06"
  ],
  "name": "Kakeya problem over finite fields",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/KSW/KSW18V47ZD.md",
  "status": null,
  "tag": "KSW18V47ZD",
  "uri": "m2://KSW18V47ZD",
  "url": "https://tamnd.github.io/math2/wiki/kakeya-problem-over-finite-fields/",
  "writing": {
    "tag": "KSW18V47ZD",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}