Some Fractal Questions In The Representation Theory Of Numbers

This dissertation mainly discusses some fractal questions arising in the representation theory of numbers.In this dissertation,we calculate the Hausdorff dimension of the excep-tional sets concerning the Erd(?)s-R(?)nyi limit theorem in?-expansion and talk about the frac-tal structures when these sets intersect with the exceptional sets in the classic Borel's normal number theorem in dyadic expansion.We also consider the distribution of the leading partial quotient and partial summation with some specific increasing speed,and investigate some relevant cases in the general Gauss-like systems.The whole dissertation is divided into six chapters,among which the related backgrounds and some preliminaries will be given in the first two chapters.We will talk about the above several aspects specifically in the following three chapters.In the third chapter,we calculate the Hausdorff dimension of the exceptional sets aris-ing in the Erd(?)s-R(?)nyi limit theorem of?-expansion.What's more,we also consider the intersections of the classic Besicovitch sets and Erd(?)s-R(?)nyi sets,and determinate its Haus-dorff dimension.Besides that,we investigate the Hausdorff dimension for the intersections of exceptional sets in the Borel's normal number theorem and Erd(?)s-R(?)nyi limit theorem.In the fourth chapter,we consider the distribution of the divergence points for the lead-ing partial quotient and partial summation with increasing speed e^n?(?>0)and calculate the Hausdorff dimension of the related exceptional sets.In the fifth chapter,we turn to the level sets concerning the big Birkhoff sums in Gauss-like systems.We obtain the Hausdorff dimension of the level sets when the boundary cases of the growth rate function for different unbounded potential functions are concerned.Finally,the last chapter is devoted to summarizing our main results in this dissertation and proposing some questions for further study.