Measure Theory Of Diophantine Approximation

The problem of Diophantine approximation is an important topic in Number Theory.The study of Diophantine approximation in manifolds is one of the most active research directions in recent years.A lot of results have been obtained by using the method of dynamic system to study the diophantine approximation of matrix.This thesis focuses on measure problems on Diophantine approximationn of matrix with dependent quantities and improvemet of Dirichlet theorem.The conditions of extremality and strongly extremality are summarized.For the improvement of the Dirichlet Theorem,it is discussed under the Lebesgue measure,friendly measure and Federer measure.It is proved that the modified Dirichlet Theorem is still true under the corresponding conditions.The results in this thesis partly generalize the conclusions of the literatures.