Constructing Simultaneous Diophantine Approximations Of Certain Cubic Numbers

Persistent Link:
http://hdl.handle.net/10150/338879
Title:
Constructing Simultaneous Diophantine Approximations Of Certain Cubic Numbers
Author:
Hinkel, Dustin
Issue Date:
2014
Publisher:
The University of Arizona.
Rights:
Copyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.
Abstract:
For K a cubic field with only one real embedding and α, β ϵ K, we show how to construct an increasing sequence {m_n} of positive integers and a subsequence {ψ_n} such that (for some constructible constants γ₁, γ₂ > 0): max{ǁm_nαǁ,ǁm_nβǁ} < [(γ₁)/(m_n^(¹/²))] and ǁψ_nαǁ < γ₂/[ψ_n^(¹/²) log ψ_n] for all n. As a consequence, we have ψ_nǁψ_nαǁǁψ_nβǁ < [(γ₁ γ₂)/(log ψ_n)] for all n, thus giving an effective proof of Littlewood's conjecture for the pair (α, β). Our proofs are elementary and use only standard results from algebraic number theory and the theory of continued fractions.
Type:
text; Electronic Dissertation
Keywords:
Cubic Numbers; Diophantine approximation; Littlewood Conjecture; Number Theory; Simultaneous Diophantine approximation; Cubic Field; Mathematics
Degree Name:
Ph.D.
Degree Level:
doctoral
Degree Program:
Graduate College; Mathematics
Degree Grantor:
University of Arizona
Advisor:
Rychlik, Marek

Full metadata record

DC FieldValue Language
dc.language.isoen_USen
dc.titleConstructing Simultaneous Diophantine Approximations Of Certain Cubic Numbersen_US
dc.creatorHinkel, Dustinen_US
dc.contributor.authorHinkel, Dustinen_US
dc.date.issued2014-
dc.publisherThe University of Arizona.en_US
dc.rightsCopyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.en_US
dc.description.abstractFor K a cubic field with only one real embedding and α, β ϵ K, we show how to construct an increasing sequence {m_n} of positive integers and a subsequence {ψ_n} such that (for some constructible constants γ₁, γ₂ > 0): max{ǁm_nαǁ,ǁm_nβǁ} < [(γ₁)/(m_n^(¹/²))] and ǁψ_nαǁ < γ₂/[ψ_n^(¹/²) log ψ_n] for all n. As a consequence, we have ψ_nǁψ_nαǁǁψ_nβǁ < [(γ₁ γ₂)/(log ψ_n)] for all n, thus giving an effective proof of Littlewood's conjecture for the pair (α, β). Our proofs are elementary and use only standard results from algebraic number theory and the theory of continued fractions.en_US
dc.typetexten
dc.typeElectronic Dissertationen
dc.subjectCubic Numbersen_US
dc.subjectDiophantine approximationen_US
dc.subjectLittlewood Conjectureen_US
dc.subjectNumber Theoryen_US
dc.subjectSimultaneous Diophantine approximationen_US
dc.subjectCubic Fielden_US
dc.subjectMathematicsen_US
thesis.degree.namePh.D.en_US
thesis.degree.leveldoctoralen_US
thesis.degree.disciplineGraduate Collegeen_US
thesis.degree.disciplineMathematicsen_US
thesis.degree.grantorUniversity of Arizonaen_US
dc.contributor.advisorRychlik, Mareken_US
dc.contributor.committeememberRychlik, Mareken_US
dc.contributor.committeememberJoshi, Kirtien_US
dc.contributor.committeememberMadden, Danielen_US
dc.contributor.committeememberThakur, Dineshen_US
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