The Measure And Fractal Properties Of Dirichlet Non-Improvable Sets

Dirichlet's theorem is a fundamental result in metric Diophantine approximation.The improvability of this theorem was first considered by Davenport and Schmidt.After them,Kleinbock and Wadleigh proposed the concept of Dirichlet improvable point formally in2018 and launched relevant work.Their results show that the improvability of Dirichlet's theorem is concerned with the growth of the product of the partial quotients.This dissertation focuses on the size of uniformly Dirichlet non–improvable set,the size of exact Dirichlet non–improvable set and metric properties of the product of the partial quotients in continued fractions.This dissertation is divided into six chapter.The first two chapters introduce the research background and some preliminaries.In the next three chapters,we discuss the above three aspects in detail.First of all,we consider the ultimate growth rate of {a_n?x?a_n+1?x?: n ? 1} relative to{q_n?x?: n ? 1}.We give the definition of the uniformly Dirichlet non–improvable set and calculate its Hausdorff dimension.We also extend our result to the case of the product of a limited number of partial quotients.Afterwards,we calculate the Hausdorff dimension of the exact Dirichlet non–improvable set,i.e.,the set of points which satisfy lim sup from n=1 to ?((log(a_n?x?a_n+1?x?)/?log q_n?x??)= ?,where? is a nonnegative number.In the next part,we establish a complete characterization on the metric properties of the product of the partial quotients,including the Lebesgue measure-theoretic result and Hausdorff dimensional result.More precisely,the size of the set of points which satisfy a_n?x?· · · a_n+m-1?x?? ??n?for infinitely many n,in the sense of Lebesgue measure and Hausdorff dimension,are given completely,where m ? 1 is an integer and ? : N ? R+is a positive function.If ? is a positive non–decreasing function,we also compute the Lebesgue measure of the set of points which satisfy a_n?x?· · · a_n+m-1?x?? ??qn?x??for infinitely many n.In the end,we summarize the main results of this dissertation and propose some questions for further study.