Diophantine Approximation On Rational Plane

The measure theory of Diophantine approximation is an active research branch in the-ory of numbers in recent years.Lots of results have obtained by use of method of dynamic system,especially in terms of extreme and strong extreme of measure.In this thesis,we discuss the Dirichlet's theorem of Diophantine approximation in ho-mogeneous space.Applying method of dynamical system,we transfer the Diophantine ap-proximation of matrix into Diophantine approximation of vectors and use of the relationship between Diophantine approximation of numbers and unipotent flows on homogeneous s-paces.The ?-improved Dirichlet's theorem is obtained.We also consider the Diophantine approximation on rational plane.For the value range of the Diophantine index on rational plane,the previous discussions only limit the dual and simultaneous theory of Diophantine approximation and do not give the non-homogeneous approximation on d-dimensional rational plane.In this paper,we introduce the Pliicker co-ordinate system and the Khintchine transfer principle.Using iterative upper(lower)transfer inequality,the Diophantine approximation on d-dimensional homogeneous space is extend-ed to d-dimensional non-homogeneous space.The range of value of the Diophantine index co is determined.By use of the Diophantine index,the definition of d-extreme is given,and the relevant conclusions in dual theory are extended to the d-extreme case.In last chapter of this thesis,we discuss the Khintchine-Groshev problem under the Hausdorff measure.By constructing the dimensional function and adding some constraints,the monotonic condition of approximation function can be removed.zero-one rule in ho-mogeneous space is given.However,after the constraints are removed,the monotonicity of the approximation function cannot be removed,and the zero-one rule does not apply in non-homogeneous spaces.