{
  "id": "2406.17544",
  "version": "v1",
  "published": "2024-06-25T13:30:38.000Z",
  "updated": "2024-06-25T13:30:38.000Z",
  "title": "Diophantine approximation with a quaternary problem",
  "authors": [
    "Alessandro Gambini"
  ],
  "comment": "arXiv admin note: text overlap with arXiv:1703.02381",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $1<k<7/6$, $\\lambda_1,\\lambda_2,\\lambda_3$ and $\\lambda_4$ be non-zero real numbers, not all of the same sign such that $\\lambda_1/\\lambda_2$ is irrational and let $\\omega$ be a real number. We prove that the inequality $|\\lambda_1p_1^2+\\lambda_2p_2^2+\\lambda_3p_3^2+\\lambda_4p_4^k-\\omega|\\le (\\max_j p_j)^{-\\frac{7-6k}{14k}+\\varepsilon}$ has infinitely many solutions in prime variables $p_1,p_2,p_3,p_4$ for any $\\varepsilon>0$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-06-25T13:30:38.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "quaternary problem",
      "diophantine approximation",
      "non-zero real numbers",
      "prime variables"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}