Diophantine Approximation Of Low-level Matrix With Dependent Quantities

Diophantine approximation is one of the classical problems in the Number Theory. Recently, the new branch called Diophantine approximation with dependent quantities is developed. It involves measures of Diophantine approximation. This article focuses on Diophantine approximation properties of low-level matrices with dependent quantities. We transferred the Diophantine approximation of matrix into Diophantine approximation of vectors and used of the relationship between properties of Diophantine approximation of numbers and unipotent flows on homogeneous spaces. It explored into the conditions of extremal and strongly extremal for matrices with dependent quantities. Particularly, we studied the strongly extermal and extermal of matrices M2,2and M2,3.We characterized its extermal of the matrix by the nonplanar and (C, a)-good of the corresponding functions.For2×3matrices, we got the extremality and strong extremality under the condition of partly planar for corresponded vectors. On condition of (d。F,v) is(C,α)-good, there is a two ordered submatrix P of M2,3, of which as long as [doP,v) is nonplanar, M73is extermal. The result is an improvement of the judgment of extremality for the general matrices. At the same time, we pointed that the condition that [d。F,v) is nonplanar cannot be reduced of the judgment that M2,3is strong extremality within the method of Dynamical properties of lattices. Therefore, it is not easy to improve the judgment of the strong extremality of functional matrices M2,3.For2x2matrices, we got an improvement of (C, a)-good for functions. A definition of (C, a)-good matrix is given. The condition(d。F,v) is (C,α)-good would be replaced by that functional matrix is (C,α)-good. Finally, we got a judgment of (strong) extremality of a special kind matrix within matrix M2,2under partly planar condition. That is a method to judge whether or not M2,2is (stong) extermal whendet(M2,2)=0.