Measure Theory Of Diophantine Approximation Of Matrices With Dependent Quantities

Diophantine approximation is one of classic problems in Number Theory. It originated from the research of approximating real number by rational numbers. Recently it is developed to manifold and has formed a new branch called Diophantine approximation with dependent quantities. It involves measures of Diophantine approximation. A lot of results have already obtained by use of methods in dynamical system to study vectors and matrices with parameters. The research of inhomogeneous Diophantine approximation is more challenging and has been studied by researchers recently.This thesis focuses on homogeneous and inhomogeneous Diophantine approximation of matrix with dependent quantities. For homogeneous Diophantine approximation of matrix with dependent quantities, we transfer the Diophantine approximation of matrix into Diophantine approximation of vectors and use of the relationship between properties of Diophantine approximation of numbers and unipotent flows on homogeneous spaces. It explores into the conditions of extremality and strongly extremality for matrices with dependent quantities. Particularly, we study the strongly extremality and extremality of square matrices. We obtain that the strongly extremality of square matrices of order 3 and extremality of square matrices of any order are independent on their determinants. For inhomogeneous Diophantine approximation of matrix with dependent quantities, we use the fact that Friendly measure is strongly contracting measure to obtain an inhomogeneous strongly extremality, which is defined by a K-bi-Lipschitz function.The results in this thesis partly generalize the conclusions of the literatures.