Mass Transference Principle And The Duffin-Schaeffer Conjecture

It is a long time for studying Duffin-Schaeffer conjecture, as early as the1940s, Duffinand Schaffer had drawn a conclusion about a Lebesgue measure version of the Duffin-Schaeffer conjecture under special conditions.Then,the general discussion about Duffin-Schaeffer conjecture kicked off. The purpose of the paper is to give a Hausdorff measureversion of the Duffin-Schaeffer conjecture contrasting with the Duffin-Schaefferconjecture for Lebesgue measure, then give a complete proof by the mass transferenceprinciple.The main result of the paper is a Mass Transference Principle which allows us totransfer the Lebesgue theory of limsup sets to the general Hausdorff theory.This is rathersurprising since the latter theory is viewed to be a subtle refinement of the former. Thefirst chapter is introduction, mainly introduces the research background and itssignificance,and outlines the research status and the relevant conclusions on this issue indomestic and foreign, the structure and arrangement of the paper is also in this chapter.Inchapter2,we introduce some relevant prior knowledge,mainly including the definition ofHausdorff measure and related properties, mass distribution principle,5r covering lemma,positive full measure sets and other related concepts.In the next chapter, We will introduceMass Transference Principle on the basis of the preceding, and using Mass TransferencePrinciple to prove Duffin-Schaeffer Conjecture.In chapter4,we mainly introduce thedefinition of theK G,Bcovering lemma,and give its rigorous proof.In chapter5, we mainlyconstruct Cantor set to prove Mass Transference Principle.In the final chapter,we mainly discuss the general Mass Transference Principle and the Duffin-Schaeffer revisited.