Inhomogeneous Diophantine Approximation Of Groshev Type

Diophantine approximation is one of the important branches of Number Theory, and it has extremely widespread application in other mathematical fields, such as Functional Theory, Combinatorial Mathematics, Computational Mathematics and so on. The most basically rational approximation and Diophantine approximation on manifolds which now is very active and both attracted great attention from plenty of scholars. Especially in recent years, the thought of dynamical systems has applied in Diophantine approximation on manifolds and important developments are acheived.This paper is based on Khintchine theorem, Groshev theorem and the measure and dimension theorems for non-degenerate manifolds. The inhomogeneous Diophantine approximation of Groshev type on manifold is studied. Major work is to discuss the inhomogeneous convergent theory of Diophantine approximation restricted to non-degenerate manifold M in R", based on the proof of Barker-Sprindzuk conjecture, the homogeneous theory of Diophantine approximation and the inhomogeneous Groshev type theory for Diophantine approximation, by the decomposition of the set in manifold, with the aid of Borel Cantell lemma and the transformation of (C, α)-good lemma and its properties and the main inhomogeneous conversion principle, we know these two types of sets in sense of Lebesgue measure is zero provided that the convergent sum condition is satisfied, from which several conclusions about the inhomogeneous convergent thoery of Diophantine approximation are obtained. The main result is that Lebesgue measure is inhomogeneous strongly extremal. At last we use the fact that friendly measure is strongly contracting measure to develop an inhomogeneous strong extreme measures which is restricted to matrices with dependent quantities.The results in this paper are partly extended and improved the existed theorems.