Fractals In The Representation Theory Of Real Numbers

The representation theory of real numbers is closely related to Diophantine approximation theory and fractal theory.In this dissertation,we study the following problems:the relative growth properties of the digits and the Hausdorff dimensions of the Jarník-like sets in the first Ostrogradsky expansions and Sylvester expansions,and the metric properties of the exact approximation sets in the Liiroth expansions.We describe the size of a set mainly from the point of metric view as well as the Hausdorff dimension.There are six chapters in this paper.In the first two chapters,the research background and some basic knowledge of the problems are introduced,and the following three chapters will concretely discuss the above problems.In the third chapter,we study the relative growth rates of the digits in the first Ostrogradsky expansions,i.e.,the exceptional sets:{x(?)[0,1):q_n(x)? ?(n) for infinitely many n(?N}and{x(?)[0,1):q_n(x)??(n),n(?) N},where ? is a positive function defined on N,?q_n(x)}_n?1 is the digit sequence with respect to the first Ostrogradsky expansions of x.Denoting by(?)the corresponding convergents,we also study the Jarník-like set of real numbers which can be well approximated by infinitely many convergents:(?).In the fourth chapter,we consider the set defined in terms of the growth rates of digits in the Sylvester expansions,i.e.,(?)where ? is a positive function defined on N,?d_n(x)}_n?1 is the digit sequence with respect to the Sylvester expansions of x.We obtain that the Hausdorff dimension of the set E(?)is lim in (?) when E(?)\{1}?(?).We also consider the set of points with (?).In addition,we prove that the Hausdorff dimension of the Jarník-like set (?) is 1/v+1,where (?) is the corresponding sequence of convergents.In the fifth chapter,we study the metric properties of the exact approximation set F(?)={x(?)(0,1]:d_n+1(x)?q_n^?(x)for infinitely many n (?)N?which is defined in terms of the digits ?d_n(x)}_n?1 and denominators ?q_n(x)?_n?1 of convergents of the Liiroth expansions,i.e.,the Lebesgue measure and Hausdorff dimension of the set F(?) are calculated.Moreover,we obtain the Hausdorff dimensions of the sets (? )and (?)The last chapter summarizes the main research results of this paper and we put forward the problems that need to be further studied.