{
  "id": "1504.04984",
  "version": "v1",
  "published": "2015-04-20T09:27:18.000Z",
  "updated": "2015-04-20T09:27:18.000Z",
  "title": "Describability via ubiquity and eutaxy in Diophantine approximation",
  "authors": [
    "Arnaud Durand"
  ],
  "comment": "94 pages. Notes based on lectures given during the 2012 Program on Stochastics, Dimension and Dynamics at Morningside Center of Mathematics, the 2013 Arithmetic Geometry Year at Poncelet Laboratory, and the 2014 Spring School in Analysis held at Universite Blaise Pascal",
  "categories": [
    "math.CA",
    "math.NT",
    "math.PR"
  ],
  "abstract": "We present a comprehensive framework for the study of the size and large intersection properties of sets of limsup type that arise naturally in Diophantine approximation and multifractal analysis. This setting encompasses the classical ubiquity techniques, as well as the mass and the large intersection transference principles, thereby leading to a thorough description of the properties in terms of Hausdorff measures and large intersection classes associated with general gauge functions. The sets issued from eutaxic sequences of points and optimal regular systems may naturally be described within this framework. The discussed applications include the classical homogeneous and inhomogeneous approximation, the approximation by algebraic numbers, the approximation by fractional parts, the study of uniform and Poisson random coverings, and the multifractal analysis of L{\\'e}vy processes.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-04-20T09:27:18.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "diophantine approximation",
      "describability",
      "multifractal analysis",
      "large intersection transference principles",
      "poisson random coverings"
    ],
    "tags": [
      "lecture notes"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 94,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150404984D"
    }
  }
}