Lévy Constants Of Continued Fractions And Dimension Of Certain Set

The continued fraction theory is closely related to the Diophantine approximation theory.Using continued fraction as a tool to represent irrational numbers is a common method to study the approximation properties of irrational numbers.There is a unique infinite continued fraction expansion:(?) for any x?R/Q.In this paper,we mainly study Lévy constants of continued fractions and dimension of certain set.Let (?) are convergents.By the properties of denominator of convergent,we know any n?2,qn(x)?2(n-1)/2;and there is an absolute constant B>0,we have qn(x)?eBn for some sufficiently large number n.It can be seen from the above that for almost all x,there are two constants a and b greater than 1,we have(?)for some sufficiently large number n.There is an absolute constant r,such that(?)as n tends to infinite for almost everywhere.Which Khintchine have proofed in 1935.Let(?) If ?*(x)=?*(x),then we said the Lévy constants of x exists,and note it as ?(x).Lévy has proofed:?(x)=?2/12log2 for almost all of x.Obviously,The Lebesgue measure of the set of numbers whose Levy constants does not exist or exists but is not equal to ?2/12log2 is 0.Based on this,we will proof:If(?).Then the Hausdorff dimension of set A is 0.This paper is divided into the following four chapters:The chapter ? is introduction,which mainly introduces the research background,the research status and the structure of this paper.In chapter ?,we recall the basic knowledge of the definition and properties of continued fraction,Hausdorff measure and Hausdorff dimension,etc.The chapter ? is the main part of this paper,we will learn the relationship between the growth rate for convergents denominator of infinite continued fraction and the summation of partial quotient.By using the Stirling formula and by the way of finding a countable cover of set A,we will proof the random s-dimension Hausdorff measure of the set A is finite,then we can proof the Hausdorff dimension of set A is 0.Finally,the chapter ? will focus on our conclusions.