On The Distribution For The Trimmed Sums Of Partial Quotients In Continued Fraction Expansions

In order to represent real number purely in mathematic, people began to research con-tinued fractions. As early as300BC, When Eulcid studied greatest common divisor, itproduced one by-product continued fractions. Continued fractions theory researches onespecial algorithm, it is one of most important application tools in computer、probabilitytheory、mathematics analysis、mechanics, especially in number theory. Furthermore, inthe representation of real numbers, at least in principle, continued fractions play a role likethe decimal in decimal system or other general system.We all know, real numbers include ra-tional and irrational numbers.A rational number can be showed by finite continued fraction.For all the rational number,it can be said by finite continued fraction in two different forms.Similarly, irrational number are also can be represented by infinite continued fractions, andit has only one accurate representation.Let x∈[0,1) and [a₁（x）, a₂（x）,···] be the continued fraction expansion of x. For anyn≥1, write Sn（x）=∑_kⁿ=1ak（x）. In1935, Khintchine proved thatSn（x）/n log nconverges inmeasure to1/log2with respect to the one dimensioal Lebesgue measure. Philipp proved thatthe partial quotients {an（x）}n≥1cannot satisfy a strong law of numbers for any reasonablygrowing norming sequence. However, Diamond and Vaaler showed that the partial quotientssatisfy a strong law of large numbers with normalizing sequence {n log n}n≥1if the largestone is trimmed. Wu and Xu proved that the Hausdorff dimension of the exceptional set is1.In this thesis, we consider the sets of reals whose trimmed sums of partial quotients tend toinfinity polynomially. The Hausdorff dimensions of such sets are determined.This thesis is divided into four parts. We give a brief overview of associated back-ground and current situation in the introductory chapter. In Chapter2, we give some relateddefinitions, elementary properties and relevant conclusions. The third part is the maincontent of this paper. We present some improvements and comments in the final of thisthesis.