Some Fractal Problems Arising In Diophantine Approximation

In this dissertation, we are concerned with the Hausdorff dimension of a kind of sets in continued fractions whose partial quotients obey some relative conditions. We study the localized uniformly Jarnik set and determine its Hausdorff dimension. Also we discuss some problems in Diophantine approximation with restricted denominators and give some sufficient conditions for quasi Ahlfors-David regularity of Moran set and quasi-Lipschitz equivalence of Moran sets.The first chapter describes the background of this paper, and some preliminaries are given in chapter2. Then with four chapters, the four issues are discussed in detail.In chapter3, for any β>0,we determine the Hausdorff dimension of the setIn chapter4, we call the set a localized uniformly Jarnik set, where Ï„:[0,1]â†'(0,+∞) is a continuous function, and determine the Hausdorff dimension of Uloc(Ï„).In chapter5, we give an affirmative answer to F. Adiceam’s question in Diophantine approximation with restricted denominators and prove a stronger result.In chapter6, we obtain some sufficient conditions for quasi Ahlfors-David regularity of Moran set and quasi-Lipschitz equivalence of Moran sets.Finally, in the seventh chapter, we summarize the main results of this dissertation, and list some topics for futher research.