{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $G$ be a finite [[abelian group|082GHNPJ53]], let $f \\colon G \\to \\C$ be a function, and let $k \\ge 1$ be an integer.\nFor $\\omega = (\\omega_1, \\dots, \\omega_k) \\in \\set{0, 1}^k$ and $h = (h_1, \\dots, h_k) \\in G^k$, write $\\omega \\cdot h = \\omega_1 h_1 + \\dots + \\omega_k h_k$ and $\\abs{\\omega} = \\omega_1 + \\dots + \\omega_k$.\nLet $\\mathcal{C}$ be [[complex conjugation|0FTH2E9TAS]], so $\\mathcal{C}^m z$ is $z$ for even $m$ and $\\bar z$ for odd $m$.\nThe *Gowers $U^k$ norm* of $f$ is the nonnegative real number $\\norm{f}_{U^k}$ with\n$$\\norm{f}_{U^k}^{2^k} = \\frac{1}{\\abs{G}^{k+1}} \\sum_{x \\in G} \\sum_{h \\in G^k} \\prod_{\\omega \\in \\set{0,1}^k} \\mathcal{C}^{\\abs{\\omega}} f(x + \\omega \\cdot h).$$\n\n## Remarks\n\nThe sum on the right is a nonnegative real number, so the root is defined.\nFor $k \\ge 2$ the map $f \\mapsto \\norm{f}_{U^k}$ is a [[norm|0V2WP9B1MF]] on the space of functions $G \\to \\C$.\nFor $k = 2$, with the Fourier transform $\\hat f(\\chi) = \\abs{G}^{-1} \\sum_x f(x) \\overline{\\chi(x)}$, one has $\\norm{f}_{U^2}^4 = \\sum_\\chi \\abs{\\hat f(\\chi)}^4$.\nFor a character $\\chi$ of $G$, each product $\\chi(x) \\overline{\\chi(x + h_1)} \\, \\overline{\\chi(x + h_2)} \\, \\chi(x + h_1 + h_2)$ equals $1$, so $\\norm{\\chi}_{U^2} = 1$.\nFor $k = 1$ the formula gives $\\norm{f}_{U^1} = \\abs{\\abs{G}^{-1} \\sum_x f(x)}$, which is only a seminorm.\nA nontrivial character $\\chi$ has $\\norm{\\chi}_{U^1} = 0$ but $\\chi \\ne 0$, so $U^1$ fails positive definiteness.\nThe norms increase with $k$: $\\norm{f}_{U^k} \\le \\norm{f}_{U^{k+1}}$.\n",
  "commit": "5fc425f97f1f6b11da5892c2948ef8c5d8805665",
  "front": {
    "defines": "ZBFZGA1J5G",
    "kind": "definition",
    "msc": [
      "11B30",
      "05D10"
    ],
    "name": "Gowers norm",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Gowers_norm"
      }
    ],
    "tag": "J3JV10S9YX",
    "uses": [
      "082GHNPJ53",
      "0FTH2E9TAS",
      "0V2WP9B1MF"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/J3JV10S9YX/",
  "links": [
    {
      "field": "defines",
      "target": "ZBFZGA1J5G"
    },
    {
      "field": "uses",
      "target": "082GHNPJ53"
    },
    {
      "field": "uses",
      "target": "0FTH2E9TAS"
    },
    {
      "field": "uses",
      "target": "0V2WP9B1MF"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/J3JV10S9YX.md",
  "msc": [
    "11B30",
    "05D10"
  ],
  "name": "Gowers norm",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/J3J/J3JV10S9YX.md",
  "status": null,
  "tag": "J3JV10S9YX",
  "uri": "m2://J3JV10S9YX",
  "url": "https://math2.tamnd.com/wiki/gowers-norm-definition/",
  "writing": {
    "tag": "J3JV10S9YX",
    "kind": "definition",
    "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
      }
    ]
  }
}