The Non-homogeneous Groshev Convergence theorem for Diophantine Approximation on Manifolds

Faiza Akram

doi:10.24297/jam.v13i4.6352

Abstract

This paper is based on Khintchine theorem, Groshev theorem and measure and dimension theorems for non-degenerate manifolds. The inhomogeneous Diophantine approximation of Groshev type on manifolds is studied. Major work is to discuss the inhomogeneous convergent theory of Diophantine approximation restricted to non-degenerate manifold in , based on the proof of Barker-Sprindzuk conjecture, the homogeneous theory of Diophantine approximation and inhomogeneous Groshev type theory for Diophantine approximation, by the decomposition of the set in manifold, with the aid of Borel Cantell lemma and transformation of lemma and its properties and the main inhomogeneous conversion principle, we know these two types of set in sense of Lebesgue measure is zero provided that the convergent sum condition is satisfied, from which several conclusions about the inhomogeneous convergent theory of Diophantine approximation is 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 measure which is restricted to matrices with dependent quantities

Highlights

The idea of rational approach was long ago, People often use a rational approximation and approximation to represent a certain number, the development and on the basis of Diophantine approximation (Diophantine Approximation) is a historical theory a long and important branch, For example, we are familiar with the history of Pie, is the rational approximation of irrational numbers π, the real relatively systematic approximation theory was developed in 19th Century with the establishment of the theory of real numbers, it has become one of the most active branches of number theory
The development of Diophantine approximation theory can be broadly divided into two major categories, Respectively, The Diophantine approximation in primitive number theory and the Diophantine approximation on Manifolds, Among them, the former research history is relatively long, the main research questions are: (1) Approximation of a single real number; (2) Simultaneous approximation of multiple real numbers; (3) Homogeneous approximation and nonhomogeneous approximation; (4 ) Rational approximation of algebraic numbers; ( 5 ) Uniform distribution of Diophantine approximation; ( 6 ) Metric theorem; ( 7 ) Uniform sequence distribution; ( 8 ) ρ - adic Diophantine approximation, etc, See literature [1] for details
This paper mainly studies the manifold dual Diophantine approximation non-homogeneous theory problem, namely homogeneous dual Diophantine approximation of Khintchine - Groshev theorem to the non-homogeneous dual

Summary

INTRODUCTION

The idea of rational approach was long ago, People often use a rational approximation and approximation to represent a certain number, the development and on the basis of Diophantine approximation (Diophantine Approximation) is a historical theory a long and important branch, For example, we are familiar with the history of Pie, is the rational approximation of irrational numbers π, the real relatively systematic approximation theory was developed in 19th Century with the establishment of the theory of real numbers, it has become one of the most active branches of number theory. The development of Diophantine approximation theory can be broadly divided into two major categories, Respectively, The Diophantine approximation in primitive number theory and the Diophantine approximation on Manifolds, Among them, the former research history is relatively long, the main research questions are: (1) Approximation of a single real number; (2) Simultaneous approximation of multiple real numbers; (3) Homogeneous approximation and nonhomogeneous approximation; (4 ) Rational approximation of algebraic numbers; ( 5 ) Uniform distribution of Diophantine approximation; ( 6 ) Metric theorem; ( 7 ) Uniform sequence distribution; ( 8 ) ρ - adic Diophantine approximation, etc, See literature [1] for details These are the classical results of linear Diophantine approximation. With the help of friendly measure is a strong contraction at the end of this paper, further promotion of a class of non homogeneous strongly extreme measure with variable matrix space

Definition of Diophantine approximation and its symbolic representation

Proof of theorem

Conclusion