{
  "id": "2312.06110",
  "version": "v1",
  "published": "2023-12-11T04:29:30.000Z",
  "updated": "2023-12-11T04:29:30.000Z",
  "title": "The exceptional set for Diophantine approximation with mixed powers of prime variables",
  "authors": [
    "Yuhui Liu"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Let lambda_1, \\lambda_2, \\lambda_3, \\lambda_4 be non-zero real numbers, not all negative, with \\lambda_1/\\lambda_2 irrational and algebraic. Suppose that \\mathcal{V} is a well-spaced sequence and \\delta >0. In this paper, it is proved that for any \\varepsilon >0, the number of v \\in \\mathcal{V} with v \\leqslant N for which |\\lambda_1 p_1^2 + \\lambda_2 p_2^3+ \\lambda_3 p_3^4+ \\lambda_4 p_4^5 - v| < v^{-\\delta} has no solution in prime variables p_1,p_2,p_3,p_4 does not exceed O\\big(N^{\\frac{359}{378} + 2\\delta +\\varepsilon}\\big). This result constitutes an improvement upon that of Q. W. Mu and Z. P. Gao [12].",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-12-11T04:29:30.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "prime variables",
      "exceptional set",
      "diophantine approximation",
      "mixed powers",
      "non-zero real numbers"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}