{
  "id": "1705.06801",
  "version": "v1",
  "published": "2017-05-18T21:16:37.000Z",
  "updated": "2017-05-18T21:16:37.000Z",
  "title": "Good bounds in certain systems of true complexity $1$",
  "authors": [
    "Freddie Manners"
  ],
  "comment": "40 pages",
  "categories": [
    "math.NT",
    "math.CO"
  ],
  "abstract": "Let $\\Phi = (\\phi_1,\\dots,\\phi_6)$ be a system of $6$ linear forms in $3$ variables, i.e. $\\phi_i \\colon \\mathbb{Z}^3 \\to \\mathbb{Z}$ for each $i$. Suppose also that $\\Phi$ has Cauchy--Schwarz complexity $2$ and true complexity $1$, in the sense defined by Gowers and Wolf; in fact this is true generically in this setting. Finally let $G = \\mathbb{F}_p^n$ for any $p$ prime and $n \\ge 1$. Then we show that multilinear averages by $\\Phi$ are controlled by the $U^2$-norm, with a polynomial dependence; i.e. if $f_1,\\dots,f_6 \\colon G \\to \\mathbb{C}$ are functions with $\\|f_i\\|_{\\infty} \\le 1$ for each $i$, then for each $j$, $1 \\le j \\le 6$: \\[ \\left| \\mathbb{E}_{x_1,x_2,x_3 \\in G} f_1(\\varphi_1(x_1,x_2,x_3)) \\dots f_6(\\phi_6(x_1,x_2,x_3)) \\right| \\le \\|f_j\\|_{U^2}^{1/C} \\] for some $C > 0$ depending on $\\Phi$. This recovers and strengthens a result of Gowers and Wolf in these cases. Moreover, the proof uses only multiple applications of the Cauchy--Schwarz inequality, avoiding appeals to the inverse theory of the Gowers norms. We also show that some dependence of $C$ on $\\Phi$ is necessary; that is, the constant $C$ can unavoidably become large as the coefficients of $\\Phi$ grow.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-05-18T21:16:37.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "true complexity",
      "linear forms",
      "inverse theory",
      "cauchy-schwarz inequality",
      "multiple applications"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 40,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}