Unique Expansions Of Real Numbers And β-dynamical System

This article focuses on the sizes of the univoque sets of real numbers, inhomogeneous Diophantine approximation in β-dynamical system and the Diophantine properties of real numbers in different β-dynamical systems. We calculate the Lebesgue measures and the Hausdorff dimensions of the related fractal sets. The article is divided into six chapter:In the first chapter, we introduce the history of fractal geometry and the backgrounds. The second chapter is devoted to giving the definition and properties of Hausdorff dimension and the preliminaries. Then, in the following three chapters, we discuss the three issues in detail.In the third chapter, for any real number x∈[0,1], we investigate the size of the univoque set, i.e. the set of β∈(1,2) such that x has a unique expansion in base β. We find that for all x∈(0,1):(1) the number2is an accumulation point of the univoque set;(2) the univoque set is a Lebesgue null set;(3) the univoque set is of Hausdorff dimension1. Furthermore, together with the result of de Vries and Komornik, we obtain that for any x∈(0,1), the univoque set and its closure are nowhere dense.In the fourth chapter, for any real number β>1, we consider the inhomogeneous Dio-phantine approximation in β-dynamical system. More precisely, for any positive function φ:Nâ†'R+, we determine the Lebesgue measure and the Hausdorff dimension of the following setEβ(φ)={(x,y)∈[0,1]×[0,1]:|Tmβx-y<φp(n) for infinitely many n∈N}, and get the following results:(1) the Lebesgue measure of the set Eβ(φ) satisfies the so called0-1law, i.e., if the series Σφ(n) converges, then its two-dimensional Lebesgue mea-sure is0; otherwise, its two-dimensional Lebesgue measure equals1.(2) the Hausdorff dimension of the set Eβ(φ) is1/(1+α), where α=lim inf(?)In the fifth chapter, for any real number x∈(0,1), we study the set of β>1such that the orbit of x under the β-transformation has a good approximation property. More precisely, for any sequence {xn} of real numbers and any positive function φ:Nâ†'R+, using mass transference principle, we prove that the Hausdorff dimension of the set Ex({xn},φ)={β>1:|Tmβx-xn|<φ(n) for infinitely many n∈N} satisfies the so called0-1law, i.e., if then its Hausdorff dimension is0; otherwise, its Hausdorff dimension is1. With the same method, we generalize the results of Persson and Schmeling[2] and Li et al.[3].In the last chapter, we first summarize the main results of this article. Then, for each topic considered, we list some questions for further study.