{
  "id": "2111.03838",
  "version": "v2",
  "published": "2021-11-06T09:09:12.000Z",
  "updated": "2022-12-01T08:57:50.000Z",
  "title": "New bound for Roth's theorem with generalized coefficients",
  "authors": [
    "Cédric Pilatte"
  ],
  "comment": "21 pages, 1 figure",
  "journal": "Discrete Analysis, 2022:16, 21 pp",
  "doi": "10.19086/da.55553",
  "categories": [
    "math.CO",
    "math.NT"
  ],
  "abstract": "We prove the following conjecture of Shkredov and Solymosi: every subset $A \\subset \\mathbf{Z}^2$ such that $\\sum_{a\\in A\\setminus\\{0\\}} 1/\\left\\|a\\right\\|^{2} = +\\infty$ contains the three vertices of an isosceles right triangle. To do this, we adapt the proof of the recent breakthrough by Bloom and Sisask on sets without three-term arithmetic progressions, to handle more general equations of the form $T_1a_1+T_2a_2+T_3a_3 = 0$ in a finite abelian group $G$, where the $T_i$'s are automorphisms of $G$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2022-12-01T08:57:50.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "roths theorem",
      "generalized coefficients",
      "isosceles right triangle",
      "three-term arithmetic progressions",
      "finite abelian group"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}