Multiplicative Problems In Diophantine Approximation

Diophantine approximation is a branch of number theory,its core issues is to quantitatively analyse approximation to real numbers by rationals.In the past few decades,this branch has made a lot of remarkable achievements.For example,Khintchine theorem,Jarn(?)k theorem and the Duffin-Schaeffer conjecture have been solved by koukoulopoulos and Maynard.In higher dimensions,Diophantine approximation problems can be divided into three types: simultaneous,dual and multiplicative.In this paper,we focus on multiplicative Diophantine approximation problems.The theory of simultaneous and dual Diophantine approximation have been very perfect,while the theory of multiplicative approximation on matrices has not been completely established.Firstly,we consider multiplicative Diophantine approximation on matrix approximation,and complete the Hausdorff measure version of the matrix form of Gallagher's theorem in the inhomogeneous setting.Subsequently,we restrict the coordinates of the approximated points by functions and consider the Diophantine approximation on manifolds.The milestone conclusion of Diophantine approximation on manifold is the Baker-Sprind(?)uk conjecture solved by Kleinbock and Margulis in 1998.It proved that almost all points on any non-degenerate manifoldMcannot be well simultaneous approximation.Over the last decade or so,the theory of Diophantine approximation has developed rapidly.Lots of the works about Diophantine approximation on manifolds ask the manifolds should be nondegenerate.For the degenerate case,the known results are still relatively few.Therefore,we consider the multiplicative Diophantine approximation on lines.There are six chapters in this paper.The first chapter mainly introduces the relevant backgrounds,development status and our main results of this paper.The second chapter mainly introduces some preliminaries.In the third chapter,we consider the multiplicative Diophantine approximation problem on matrix approximation.Let n,m?1 be two integers,y =(y₁,y₂,...,y_n)(?)[0,1]ⁿ is a fixed point.Suppose that ? : Z^m?R⁺is a multivariate approximating function,let (?).When m = 1,Beresevich and Velani,Hussain and Simmons obtained the sdimensional Hausdorff measure theory of the set M_n,1^y.In addition,Hussain and Simmons posed a conjecture about the s-dimensional Hausdorff measure on M_n,m^y(?).Therefore,we obtain the Hausdorff measure theoretic results for the set M_n,m^y(?),it is either zero or infinity depending upon whether a series converges or diverges.In the fourth chapter,we consider the general case of multiplicative Diophantine approximation on matrix approximation.Let ?₁,?₂,...,?_n be positive real number with A(n)= ?₁+ ?₂+· · ·+ ?_n.Following the works of the above,we prove a dichotomy law of the s-dimensional Hausdorff measure for the set (?) and obtain the Hausdorff dimension of the set.In the fifth chapter,we consider multiplicative Diophantine approximation of the expansions under different bases on a line and obtain the s-dimensional Hausdorff measure of the following set (?) is a positive function without monotonicity.At last,we summarize the main results of this paper and introduce some problems that we are investigating and propose some Diophantine approximation problems that can be studied in the future.