【作者】 高翔；

【导师】 吴军；

【作者基本信息】 华中科技大学 ， 基础数学， 2010， 硕士

【摘要】 两个世纪以来,丢番图逼近(Diophantine Approximation)的研究取得了许多重大的进展,现已经成为数论中一个重要的分支.本文首先回顾度量数论中一些重要的结果(包括Khintchine定理和Jarnik定理).其中引出了一个重要的概念:不可很好逼近数(badly approximable numbers),并记Bad为所有的不可很好逼近数的集合,随后研究了这个集合的一些重要的度量性质.文中详细介绍了连分数的很多重要的性质,尤其研究了部分商有界的情形.紧接着,我们重点概括了齐次丢番图逼近中著名的Littlewood猜想的一些结果,并且介绍了它的最新的进展.同时还列举了几个跟Littlewood猜想紧密相关的问题和猜想.此外,本文重点介绍Hausdorff测度和维数,同时给出一些熟知的结论和概念,特别强调了维数的计算方法.最后,我们讨论了序列的模1一致分布(uniform distribution modulo one).本文主要目的是在有关Littlewood猜想的问题上给出两个基本且简单的结果.第一个,任意给定一个部分商有界的无理数α,我们构造一个具体的且部分商同样有界的实数β,使得实数对(α,β)满足Littlewood猜想.证明主要依靠的是连分数的一些基本的性质,在此基础上用递归的方式构造实数β使其满足给定的要求,最后就是加以验证.第二个结果是首先证明了存在实数对(α,β)满足下列不等式: liminf 20q→∞(q logq) qαqβ>进一步给出满足该条件的实数对(α,β)组成的集合是满Hausdorff维数的.本文的方法最先是由Peres和Schlag使用的,也就是说首先证明满足给定性质的实数是存在的,其次证明这些满足性质的数组成的集合的Hausdorff维数是满的.更多还原

【Abstract】 Since two centuries the theory of Diophantine approximation has got a tremendous development. It has been an important branch of the number theory.In this paper we recall some classical results (including Khintchine’s theorem and Jarnik’s theorem) without proof and give an important concept of badly approximable numbers, meanwhile we associates with the theory of continued fractions, especially discuss the real number with bounded partial quotients. Subsequently, we begin by a survey on the celebrated Littlewood conjecture and focus on summarizing some recent developments on this conjecture. Besides, we introduce some famous problems which are related to the Littlewood conjecture. We recall some well-known results on Hausdorff measure and dimension. Significantly we present a useful theorem which is popularly used in calculating the Hausdorff dimension. The last section is concerned with the theory of uniform distribution modulo one.The main purpose of this paper is to provide new, short and elementary results on the problem which related to Littlewood conjecture. The first result is, for any given real numberαwith bounded partial quotients, we can construct explicitly continuum many real numbersβwith bounded partial quotients for which the pair(α,β)satisfies a strong form of the Littlewood conjecture. Our proof mainly rests on the basic theory of continued fractions. The second result is, we firstly prove the existence of real numbers(α,β)such that liminf 20q→∞(q logq) qαqβ>. Then we give a further result that the set of pairs (α,β) satisfies the above inequality has full Hausdorff dimension. Our method was firstly introduced by Peres and Schlag.更多还原