{
  "id": "2405.15400",
  "version": "v1",
  "published": "2024-05-24T09:53:50.000Z",
  "updated": "2024-05-24T09:53:50.000Z",
  "title": "Two-point polynomial patterns in subsets of positive density in $\\mathbb{R}^n$",
  "authors": [
    "Xuezhi Chen",
    "Changxing Miao"
  ],
  "comment": "18 pages,to appear in International Mathematics Research Notices",
  "doi": "10.1093/imrn/rnae108",
  "categories": [
    "math.CA",
    "math.NT"
  ],
  "abstract": "Let $\\gamma(t)=(P_1(t),\\ldots,P_n(t))$ where $P_i$ is a real polynomial with zero constant term for each $1\\leq i\\leq n$. We will show the existence of the configuration $\\{x,x+\\gamma(t)\\}$ in sets of positive density $\\epsilon$ in $[0,1]^n$ with a gap estimate $t\\geq \\delta(\\epsilon)$ when $P_i$'s are arbitrary, and in $[0,N]^n$ with a gap estimate $t\\geq \\delta(\\epsilon)N^n$ when $P_i$'s are of distinct degrees where $\\delta(\\epsilon)=\\exp\\left(-\\exp\\left(c\\epsilon^{-4}\\right)\\right)$ and $c$ only depends on $\\gamma$. To prove these two results, decay estimates of certain oscillatory integral operators and Bourgain's reduction are primarily utilised. For the first result, dimension-reducing arguments are also required to handle the linear dependency. For the second one, we will prove a stronger result instead, since then an anisotropic rescaling is allowed in the proof to eliminate the dependence of the decay estimate on $N$. And as a byproduct, using the strategy token to prove the latter case, we extend the corner-type Roth theorem previously proven by the first author and Guo.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-05-24T09:53:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42B20"
    ],
    "keywords": [
      "two-point polynomial patterns",
      "positive density",
      "gap estimate",
      "decay estimate",
      "oscillatory integral operators"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}