Dissertation > Mathematical sciences and chemical > Mathematics > Algebra,number theory, portfolio theory > Number Theory > The Diophantus analysis ( Diophantine number theory )

Cantor centralized approach can be a good measure of the nature of point sets

Author WangChunYan
Tutor WangBaoWei
School Huazhong University of Science and Technology
Course Basic mathematics
Keywords Hausdorff measure Hausdorff dimension Can be a good approximation set Cantor set Mass transfer theory
CLC O156.7
Type Master's thesis
Year 2011
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Diophantine approximation in number theory is an important branch of study , the main content is to study the real number of rational approximation .1842 years , Dirichlet first gives rational approximation of a real number of important conclusions In 1926 , Khintchine results , creating a with a measure significance , to study the relevant number theory problems, and now we call metric number theory Jarnik and Besicovitch first studied using Haudorff dimension and Hausdorff measure to characterize a given Diophantine approximation point set scale size and since fractal since forming geometric theory , more and more people discover the universality of fractal recent years , fractal sets Diophantine approximation is also becoming more and more in-depth study of fractal theory and Diophantine research point of integration . article discusses the Cantor set can be a good measure of the nature of the approach point set answered Mahler 's Cantor set K on the outside except Liouville number , whether there can be a good approach point problems, and gives this estimate the dimension of the collection of this thesis consists of four parts . introduction part mainly introduces some problems related research background , including knowledge in the preparation Hausdorff measure and dimension definitions and related properties , and introduces prove theorems and on paper limit sets Haudorff measure and dimension needed an important tool - mass transfer theory . behind two part is the main content. located sand one defined in the set of natural numbers on the positive real -valued function , denoted a: = {3n : n = 0,1,2 ...), so WA (ψ) represents the unit interval for infinitely many (p, q) ∈ Z × N satisfy | xp / q | lt; ψ (q) the set of points . in the third section depicts the complete collection of WA (ψ) in the sense of Hausdorff measure 0 - ∞ law should be pointed out that here we do not require the monotony of sand , it can be considered to be on the set WA (ψ) ∩ K, in the sense of Hausdorff measure of Duffin-Schaeffer theorem the last article we give the Cantor set K point can be a good approximation of the dimension of a direct simple estimate.

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