二次区间的分布及其性质

发布时间：2019-04-21 22:11

【摘要】：在本文中,我们简单扼要的介绍了连分数理论和正整数字符串,它们是本文进行分析、推导的有力工具。我们主要在]1,0(区间上集中讨论了实数的连分数展开式,并重点描述了有理数的两种展开式形式,另外还包括渐进分式之间的关系,值的大小与部分商位置的关系.通过连分数理论,我们自然的引入了正整数字符串的知识,并把重点放在两等长字符串的大小的判定上.在有了上述知识后,我们在(0,1]上构造了一族开区间,并记它们的并集为M.由于任意一个开区间的端点都是二次无理数,故称之为二次区间.针对该集合,我们详细讨论了二次区间之间的位置关系以及与其伪中心的联系,从而可以由二次区间生成数本身的特征来判定两个二次区间的位置关系.而极大区间的引入使我们更容易把握该集合中元素的分布规律.在严格的理论基础上,运用两种构造方法,我们得出了所有极大区间序列,并把他们的并集记为F?易得以下结论:FM?.文章中还给出了极大区间的判定定理,最后对M的补集,即??M\]1,0(进行讨论,分析该集合元素的一些特征,证明了它是不可数的,并进一步求得其Hausdorff维数为1.
[Abstract]:In this paper, we briefly introduce the continuous fraction theory and positive integer string, which are powerful tools for the analysis and derivation of this paper. In this paper, we mainly discuss the continuous fraction expansion of real numbers in the interval, and focus on describing the two kinds of expansion forms of rational numbers. In addition, we also include the relations between asymptotic fractions, the size of values and the position of partial quotient. Through the continuous fraction theory, we naturally introduce the knowledge of positive integer string and focus on the determination of the size of two equal-length strings. With the above knowledge, we construct a family of open intervals on (0, 1) and remember that their union is M. Because the endpoints of any open zone are quadratic irrational numbers, they are called quadratic intervals. For the set, we discuss in detail the position relation between the quadratic interval and its pseudo-center, so that the position relation of the quadratic interval can be determined by the characteristic of the number generated by the quadratic interval itself, and the position relation between the quadratic interval and its pseudo-center can be determined by the character of the quadratic interval generating number itself. The introduction of maximum interval makes it easier for us to grasp the distribution of elements in the set. On the basis of strict theory, by using two construction methods, we obtain all maximal interval sequences and denote their union as F? The following conclusions are readily available: FM?. Finally, the complement set of M, that is? M\] 1, 0 () is discussed, some characteristics of the set element are analyzed, it is proved that it is uncountable, and its Hausdorff dimension is 1. 1.
【学位授予单位】：华中科技大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O156

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