Fractals In Representation Of Numbers And Diophantine Approximation

Fractals arise naturally in the theory of representation of numbers and Diophantine approximation.In this dissertation,we study the recurrent sets in ?-symbolic dynamics,the Jarník-like set and uniformly Jarník-like set in the L(?)roth expansion.We describe the size of a set mainly from the point of metrical view as well as the Hausdorff dimension.This paper consists of six chapter.The first chapter statements the research background of the problems.The second chapter gives some basic knowledge which will be used to solve the problems.The third chapter gives the dimensions of the recurrent sets in ?-symbolic dynamics,which is defined as(?) where ?n(x,?)denotes the return time of a point x?[0,1]under the ?-transformation T? returns the n-th order cylinder set containing x;0?a?b??;?:N?R+ is a non-decreasing function.The fourth chapter determines the dimension of the set(?) where 0??????,the sequences {an(x),n?1} and {qn(x),n?1} are consist of the partial quotients and denominators of convergents of x in its continued fraction expansion,respectively.The fifth chapter completely characterizes the metric properties of the Jarník-like set and uniformly Jarník-like set in the L(?)roth expansion.More precisely,for a decreasing function?:N?R,we determine the Lebesgue measure of the Jarník-like set J*(?)={x?[0,1):|xQn(x)-Pn(x)|<?(Qn(x))for infinitely many n ?N}.Moreover,for ??0,we also calculate the Hausdorff dimension of the uniformly Jarník-like set#12 In addition,without the hypothesis on the monotonicity of ??we obtain the size of the set W(?)={x?[0,1):|xQn(x)-Pn(x)|<?(n)for infinitely many n ? N}.The last chapter summarizes the main results of this paper and poses some questions to be further studied.