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On numbers badly approximable by q-adic rationals

Nilsson, Johan LU (2007)
Abstract
The thesis takes as starting point diophantine approximation with focus on the area of badly approximable numbers. For the special kind of rationals, the q-adic rationals, we consider two types of approimations models, a one-sided and a two-sided model, and the sets of badly approximable numbers they give rise to. We prove with elementary methods that the Hausdorff dimension of these two sets depends continuously on a defining parameter, is constant Lebesgue almost every and self similar. Hence they are fractal sets. Moreover, we give the complete description of the intervals where the dimension remains unchanged. The methods and techniques in the proofs uses ideas form symbolic dynamics, combinatorics on words and the beta-shift.
Abstract (Swedish)
Avhandlingen tar som utgångspunkt diofantisk approximation med fokus på dåligt approximerbara tal. För specialfallet de q-adiska rationella talen betraktar vi två typer av approximationsmodeller, en ensidig och en tvåsidig modell, och de dåligt approximerbara tal de ger upphov till. Vi visar med elementära metoder att Hausdorff-dimensionen av dessa två mängder beror kontinuerligt på en definierande parameter, är konstant Lebesgue nästan överallt och likformig med sig själv. Vi ger även den fullständiga beskrivningen av de intervall där dimensionen är oförändrad. Metoder och tekniker i bevisen bygger på idéer från symbolisk dynamik, ordkombinatorik och beta-skift.
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author
supervisor
opponent
  • Professor Bugeaud, Yann, Université Louis Pasteur, Mathématique, Strasbourg, France
organization
publishing date
type
Thesis
publication status
published
subject
keywords
Number theory, Symbolic dynamics, Combinatorics on words, Badly approximable numbers, Diophantine approximation, Mathematics, Matematik, Dynamical systems
pages
98 pages
publisher
Centre for Mathematical Sciences, Lund University
defense location
Lecture room MH:C, Centre for mathematical sciences, Sölvegatan 18, Lund University Faculty of Engineering
defense date
2007-12-06 13:15:00
ISBN
978-91-628-7334-9
language
English
LU publication?
yes
id
6d35c625-4eb2-45f3-add8-63eb615e0be8 (old id 599328)
date added to LUP
2016-04-01 15:49:28
date last changed
2018-11-21 20:36:38
@phdthesis{6d35c625-4eb2-45f3-add8-63eb615e0be8,
  abstract     = {{The thesis takes as starting point diophantine approximation with focus on the area of badly approximable numbers. For the special kind of rationals, the q-adic rationals, we consider two types of approimations models, a one-sided and a two-sided model, and the sets of badly approximable numbers they give rise to. We prove with elementary methods that the Hausdorff dimension of these two sets depends continuously on a defining parameter, is constant Lebesgue almost every and self similar. Hence they are fractal sets. Moreover, we give the complete description of the intervals where the dimension remains unchanged. The methods and techniques in the proofs uses ideas form symbolic dynamics, combinatorics on words and the beta-shift.}},
  author       = {{Nilsson, Johan}},
  isbn         = {{978-91-628-7334-9}},
  keywords     = {{Number theory; Symbolic dynamics; Combinatorics on words; Badly approximable numbers; Diophantine approximation; Mathematics; Matematik; Dynamical systems}},
  language     = {{eng}},
  publisher    = {{Centre for Mathematical Sciences, Lund University}},
  school       = {{Lund University}},
  title        = {{On numbers badly approximable by q-adic rationals}},
  year         = {{2007}},
}