{
  "id": "math/0404498",
  "version": "v3",
  "published": "2004-04-27T19:49:54.000Z",
  "updated": "2015-04-20T13:10:13.000Z",
  "title": "Self-Similar Fractals and Arithmetic Dynamics",
  "authors": [
    "Arash Rastegar"
  ],
  "comment": "19 pages",
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "abstract": "The concept of self-similarity on subsets of algebraic varieties is defined by considering algebraic endomorphisms of the variety as `similarity' maps. Self-similar fractals are subsets of algebraic varieties which can be written as a finite and disjoint union of `similar' copies. Fractals provide a framework in which, one can unite some results and conjectures in Diophantine geometry. We define a well-behaved notion of dimension for self-similar fractals. We also prove a fractal version of Roth's theorem for algebraic points on a variety approximated by elements of a fractal subset. As a consequence, we get a fractal version of Siegel's theorem on finiteness of integral points on hyperbolic curves and a fractal version of Falting's theorem on Diophantine approximation on abelian varieties.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2006-09-30T07:07:37.000Z",
      "title": "Self-Similar Fractals in Arithmetic",
      "abstract": "We define a notion of self-similarity on algebraic varieties by considering algebraic endomorphisms as \"similarity\" maps. Self-similar objects are called fractals, for which we present several examples and define a notion of dimension in many different contexts. We also present a strong version of Roth's theorem for algebraic points on a variety approximated by fractal elements. Fractals provide a framework in which one can unite several important conjectures in Diophantine geometry.",
      "comment": "15 pages",
      "journal": null,
      "doi": null
    },
    {
      "version": "v3",
      "updated": "2015-04-20T13:10:13.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G35",
      "11J17"
    ],
    "keywords": [
      "self-similar fractals",
      "arithmetic",
      "strong version",
      "considering algebraic endomorphisms",
      "self-similar objects"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2004math......4498R"
    }
  }
}