Mass Transference Principle For Limsup Sets Generated By Rectangles

Beresnevich & Velani which allows us to transfer Lebesgue measure theoretic statements for limsup subsets ofkR to Hausdorff measure theoretic statements. This paper generalize the conclusion to limsup sets generated by rectangle. More precisely, let{ }1nnx3 be a sequ-ence of points in the unit cube [0,1]dwith d 31 and { }1nnr3a sequenceof positive numbers tending to zero. Under the assumption of full Lebesgue measure theoretical statement of the setThe 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 theG,BK-Lemma mass distribution principle, and two inequality lemma provide convenient for later proof. In the next chapter, in order to prove the theorem 1.2, at first, we construct a Cantor subset F?ofaW; secondly, define a suitable mass distribution m supported on F?;thirdly,estimate the Holder exponent of the measure m;and at last, we conclude the result by applying the mass distribution principle. In chapter 4, we mainly prove the theorem 1.3,We refine the Cantor set F?to a new one G?. Combined with the division method and inductive method to get the conclusion. In chapter 5, we mainly introduce the definition of Simultaneous diophantine approximation,the High dimensional Duffin-Schaeffer conjecture.and two corollaries.In the final chapter, we mainly discussed the related conclusion and promotion.