{
  "id": "math/9904047",
  "version": "v5",
  "published": "1999-04-12T02:15:20.000Z",
  "updated": "1999-05-05T02:37:46.000Z",
  "title": "Discrete versions of the Beckman-Quarles theorem",
  "authors": [
    "Apoloniusz Tyszka"
  ],
  "comment": "LaTeX209, 14 pages, textonly.tex, text&gif.ps, to appear in Aeq. Math",
  "journal": "Aequationes Mathematicae 59 (2000), pp.124-133",
  "categories": [
    "math.MG"
  ],
  "abstract": "TO APPEAR IN AEQUATIONES MATHEMATICAE - WITHOUT THEOREM 2. THEOREM 2 IS CORRECTLY PROVED IN PREVIOUS VERSIONS 1 AND 2. AUTHOR'S VERSION 3 (WITH A NEW FIGURE 6A) IS UNNECESSARY. Let F \\subseteq R denote the field of numbers which are constructible by means of ruler and compass. We prove that: (1) if x,y \\in R^n (n>1) and |x-y| is an algebraic number then there exists a finite set S(x,y) \\subseteq R^n containing x and y such that each map from S(x,y) to R^n preserving all unit distances preserves the distance between x and y; if x,y \\in F^n then we can choose S(x,y) \\subseteq F^n, (2) only algebraic distances |x-y| have the property from item (1), (3) if X1,X2,...,Xm \\in R^n (n>1) lie on some affine hyperplane then there exists a finite set L(X1,X2,...,Xm) \\subseteq R^n containing X1,X2,...,Xm such that each map from L(X1,X2,...,Xm) to R^n preserving all unit distances preserves the property that X1,X2,...,Xm lie on some affine hyperplane, (4) if J,K,L,M \\in R^n (n>1) and |JK|=|LM| (|JK|<|LM|) then there exists a finite set C(J,K,L,M) \\subseteq R^n containing J,K,L,M such that any map f:C(J,K,L,M) \\to R^n that preserves unit distance satisfies |f(J)f(K)|=|f(L)f(M)| (|f(J)f(K)|<|f(L)f(M)|).",
  "revisions": [
    {
      "version": "v5",
      "updated": "1999-05-05T02:37:46.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "51M05",
      "05C12"
    ],
    "keywords": [
      "beckman-quarles theorem",
      "discrete versions",
      "unit distances preserves",
      "finite set",
      "affine hyperplane"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "1999math......4047T"
    }
  }
}