Mass Transference Principle And The Duffin-Schaeffer Conjecture

It is a long time for studying Duffin-Schaeffer conjecture, as early as the 1940 s, Duffin and Schaffer had drawn a conclusion about a Lebesgue measure version of the DuffinSchaeffer conjecture under special conditions.Then,the general discussion about DuffinSchaeffer conjecture kicked off. The purpose of the paper is to give a Hausdorff measure version of the Duffin-Schaeffer conjecture contrasting with the Duffin-Schaeffer conjecture for Lebesgue measure, then give a complete proof by the mass transference principle.The main result of the paper is a Mass Transference Principle which allows us to transfer the Lebesgue theory of limsup sets to the general Hausdorff theory.This is rather surprising since the latter theory is viewed to be a subtle refinement of the former. The first chapter is introduction, mainly introduces the research background and its significance,and outlines the research status and the relevant conclusions on this issue in domestic and foreign, the structure and arrangement of the paper is also in this chapter.In chapter 2,we introduce some relevant prior knowledge,mainly including the definition of Hausdorff measure and related properties, mass distribution principle, 5r covering lemma,positive full measure sets and other related concepts.In the next chapter, We will introduce Mass Transference Principle on the basis of the preceding, and using Mass Transference Principle to prove Duffin-Schaeffer Conjecture.In chapter 4,we mainly introduce the definition of theG,BK covering lemma,and give its rigorous proof.In chapter 5, we mainly construct Cantor set?k to prove Mass Transference Principle.In the final chapter,we mainly discuss the general Mass Transference Principle and the Duffin-Schaeffer revisited.