On The Sets Related To The Largest Digits In Lüroth Expansions

In recent years,fractal theory has obtained many successful applications in number theory and dynamical system such as continued fraction theory,Diophan-tine approximation and various expansions of real number.Lüroth expansion can be regarded as a kind of“infinite branch”extension of decimal expansion,and it plays an important role in number theory and dynamical systems.LetL_n(x)denotes the largest digit of the first n terms in the Lüroth expansion of the point x in(0,1].Galambos concluded that for Lebesgue almost all the points x in(0,1],(?).Shen,Yu and Zhou(Int.J.Number Theory.(2014))showed that the set with the pointswhose limit of logL_n(x)/logn is any nonnegative real numbers have a full Hausdorff dimension.Based on the result of Shen,Yu and Zhou,this thesis consider the Hausdorff dimensions of a class of more elaborate exceptional sets.Our main result is an essential completeness of the result of Shen Yu and Zhou.Combining the result of Shen,Yu,Zhou and ours,the interval(0,1]can be decomposed into disjoint union of uncountably many sets with full Hausdorff dimensions.The structure of the paper is as follows:we introduce the research background of our topic,the research content and the purpose of this thesis in the first chapter.Some basic knowledge,such as Hausdorff dimension,self-similar set and Lüroth expansion are collected in the second chapter.The third chapter is devoted to the proofs of the main results.Finally,we proposes some unsolved problems related to our research topic.