{
  "id": "1605.02538",
  "version": "v1",
  "published": "2016-05-09T11:39:50.000Z",
  "updated": "2016-05-09T11:39:50.000Z",
  "title": "Diophantine approximation with improvement of the simultaneous control of the error and of the denominator",
  "authors": [
    "Abdelmadjid Boudaoud"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "In this work we proof the following theorem which is, in addition to someother lemmas, our main result:\\noindent \\textbf{theorem}. Let$\\ X=\\{ ( x\\_{1}\\text{, }%t\\_{1}) \\text{, }( x\\_{2}\\text{, }t\\_{2}) \\text{, ..., }(x\\_{n}\\text{, }t\\_{n})\\} $ be a finite part of $\\mathbb{R}\\times \\mathbb{R}^{\\ast +}$, then there exist a finite part $R$ of $\\mathbb{R}%^{\\ast +}$ such that for all $\\varepsilon > 0$ there exists $r\\in R$ such that if $0 < \\varepsilon \\leq r$ then there exist rational numbers $( \\dfrac{p\\_{i}}{q}) \\_{i=1,2,...,n}$ such that:\\{c}| x\\_{i}-\\dfrac{p\\_{i}}{q}| \\leq \\varepsilon t\\_{i} \\varepsilon q\\leq t\\_{i}|\\text{, }i=1,2,...,n\\text{.} \\tag{*}\\noindent It is clear that the condition $\\varepsilon q\\leq t\\_{i}$ for $%i=1,2,...,n$ is equivalent to $\\varepsilon q\\leq t=\\underset{i=1,2,...,n}{Min%}$ $( t\\_{i}) $.\\ Also, we have (*) for all $\\varepsilon $verifying $0 < \\varepsilon \\leq \\varepsilon \\_{0}=\\min R$.The previous theorem is the classical equivalent of the following one whichis formulated in the context of the nonstandard analysis ($[ 2] $%, $[ 5] $, $[ 6] $, $[ 8] $).\\noindent \\textbf{theorem. }For every positive infinitesimal real $\\varepsilon$, there exists an unlimited integer $q$ depending only of $\\varepsilon $, such that $\\forall ^{st}x \\in \\mathbb{R}\\ \\exists p_{x} \\in \\mathbb{Z}$ $:\\{ \\{ccc}x & = \\& \\dfrac{p_{x}}{q}+\\varepsilon \\phi \\varepsilon q \\& \\cong \\& 0.$ For this reason, to prove the nonstandard version of the main result and to get its classical version we place ourselves in the context of the nonstandard analysis.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-05-09T11:39:50.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "diophantine approximation",
      "simultaneous control",
      "denominator",
      "improvement",
      "finite part"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}