Hausdorff Dimension In Inhomogeneous Diophantine Approximation

Let ? be an irrational real number,?is a positive real number,and some sets are defined as follows:(?).Where,?x? in the above represents the distance from x to its nearest integer.If?c(x)>0,st ?p/q? Q,|x-p/q|>c(x)/q2,then x is called a badly approximable number.In this paper,we mainly discuss the Hausdorff dimension of the badly approximable set in inhomogeneous diophantine approximation.In 2018,Yann Bugeaud,Dong Han Kim,Seonhee Lim and Michal Rams studied the problem in which there is a limit for partial quotients ak of a.Their conclusion is that:if the partial quotients ak of a tends to infinity,then the Hausdorff dimension of the set Bad?(?)is 1.In this paper,we mainly discuss the lower bound of Hausdorff dimension of set Bad?(?)and get the the conclusion lim??0 dimHBad?(?)=1 when a is a badly approximable number and qk+1/qk? a>1,where a is a constant and qk is the denominator of k-th convergent of a.In addtion,the relationship among the Hausdorff dimension of Bad?(?),Bad(?)and E?(?)is discussed.This paper consists of four chapters.The first chapter of this paper mainly introduces the research background and current progress of inhomogeneous diophantine approximation and the questions and structural framework of this article.In the second chapter,we mainly introduce the related definitions and properties in continued fraction,the definitions and properties of Hausdorff measure and Hausdorff dimension in fractal geometry and the mass distribution principle.The third chapter is the main part of this paper.In this chapter,we give the theorems that we are committed to prove and the steps which consist of the construction of the subsets of the set Bad?(?),the estimation of the lower bound of Hausdorff dimension,and the relationship among Bad?(?),Bad(?)and E?(?).In the end,this paper generalizes the theorems proved.