Mass Transference Principle And Metric Properties For Continued Fractions Of Laurent Series

This dissertation mainly studies two aspects:one problem is the measure theory of Mass Transference Principle from balls to arbitrary shapes,the other problem is about the relative growth rate of the product of the norms of consecutive partial quotients in continued fraction expansions of Laurent series.This dissertation is divided into five chapters.In the first two chapters,we mainly introduce the relevant research background and basic knowledge of this dissertation.In the next two chapters,we will discuss the two main problems above in detail.In Chapter 3,following a work of Koivusalo and Rams in 2020,by a further generalization of the singular value function,we extend the Mass Transference Principle set up by Beresnevich and Velani to lim sup sets generated by open sets of arbitrary shapes.Koivusalo and Rams mainly discussed the dimensional theory of Mass Transference Principle from balls to arbitrary shapes,and we consider this problem in terms of measure theory.In Chapter 4,let F_q be a finite field with q elements and F_q((z^-1)) be the field of formal Laurent series,i.e.,(?)Define (?)then I is the valuation ideal of F_q((z^-1)).For any x ? I,let[A1(x),A2(x)},…]be its continued fraction expansion and let {P_n(x)/Q_n(x)}n?1 be the sequence of convergents of x.Given m? 1,define(?)and a lim sup set(?) where ?:N?R+ is a positive function and ?·? represents the norm of a polynomial in F_q((z^-1)).We calculate the Hausdorff dimension of D_m(?)and the Haar measure of F_m(?),in addition,we estimate the size of the set F_m(?).In the end,in the fifth chapter,we summarize the main results of this dissertation and put up some questions for further considerations.