Khintchine Type Diophantine Approximation Of Matrix With Dependent Quantities

Diophantine approximation is a branch of number theory. Recently, the research of Diophantine approximation has been developed to the manifolds, and led to the development of a new branch in approximation theory, usually referred to as "measure theory in Diophantine approximation" or "Diophantine approximation with dependent quantities". Researching the Diophantine approximation problems on manifolds by use of Dynamical system is widely used and very effective. It has obtained lots of significant results for Diophantine approximation of vectors and matrices with dependent quantities.Among the various forms of Diophantine approximation, the Khintchine-type Diophantine approximation is one of the active research content. In this paper, we study the Khintchine-type Diophantine approximation with dependent quantities. The convergence part of Khintchine Theorem for nondegenerate smooth sub-manifolds in Mm,n is obtained. For a positive decreasing function Ψ. We denote W(Ψ) the set of all the Ψ-approximable points lying on Mm,n.We will prove Mm,n is of Khintchine-type for convergence; i.e. If the following converges Then the measure of x which satisfied F(x)∈W(Ψ) is zero.The proof of the main theorem of this paper is based on the correspondence between Diophantine approximation properties of numbers and orbit properties of certain flows on homogeneous spaces. We characterize the measure of the approximation points of the matrix by talking the results on quantitative non-divergence estimates for unipotent flows on the space of lattices, and get the the Khintchine-type Diophantine approximation conclusion for matrices. Another effective tool which we use is transferring the Diophantine approximation of matrix into the problems of a certain form vector. The results of this paper partly generalize and improve some results of the existing literature.