Some Studies On The Dimension Theory In The Diophantine Approximation

The subject of metric Diophantine approximation is to study the metrical properties of numbers with special approximation properties.One of the earliest research in this area can be traced back to Dirichlet Theorem(1842),which describes the rate at which irrational numbers are approximated by rational numbers.Since then,much of the research,such as Khintchine Theorem(1924),Jarnik-Besicovitch Theorem(1928,1934)and Jarnik Theorem(1931),had dedicated to generalizing and strengthening Dirichlet Theorem.This thesis considers metric Diophantine approximation in two aspects:exact approximation order and uniform dynamical covering.In Chapter 1,we first present some important results in metric Diophantine approximation.Then,we introduce the background of exact approximation order and uniform dynamical covering,and state our main results accordingly.In Chapter 2,we introduce the definitions and some properties of Hausdorff measure,Hausdorff dimension,packing measure and packing dimension.In Chapter 3,we investigate the dimensions of sets of exact approximation order by complex rational numbers.Given a non-increasing function ψ,let ExactC(ψ)be the set of complex numbers which are approximable by complex rational numbers to orderψ but to no better order.We obtain the Hausdorff dimension and packing dimension of ExactC(ψ)when ψ(x)=o(x-2).Moreover,without the condition ψ(x)=o(x-2),we also prove that the Hausdorff dimension of ExactC(ψ)is greater than 2-τ/(1-2τ)when 0<τ=lim supx→+∞ x2ψ(x)small enough.In Chapter 4,we study the Hausdorff dimension of uniform dynamical covering sets.Let T:[0,1]→[0,1]be an expanding Markov map.Let μψ be a Gibbs measure associated with a Holder continuous potential φ.For x∈[0,1],and κ>0,let Uκ(T,x):={y∈[0,1]:(?)N>>1,(?) 1≤n≤N,such that |Tnx-y|<N-κ}denote the set of points that can be uniform approximated by the orbit of x.We find a critical value κ0 satisfying the following properties:If κ<κ0,then for μφ almost every x,Uκ,(T,x)has full Hausdorff dimension;if κ>κ0,then for μφ almost every x,the Hausdorff dimension of Uκ(T,x)is strictly less than 1 and coincides with the multifractal spectrum of μφ.