Limit Theorems For Sums Of Partial Quotients In L(?)roth Expansions And The Calculation Of Hausdorff Dimension

In fractal geometry, with the successful solution of related properties and classic problems in continued faction,people began to turn their attentions to the research of related problems in Luroth expansion.This article focuses on the limit theory for sums of partial quotients in Luroth expansions and the calculation of hausdorff dimension,and discuss a metric property of generalized continued fraction. The contribution of this paper is as follows:The first chapter describes the background and process of continued fraction and Luroth dimension, the background knowledge of generalized fraction. It also introdu-ces XinXin、J.Good、Jarnik’s research on the nature of continued fraction,especially cites Luroth and Si Kui Wang’s achievements on Luroth expansions.bases on this, the paper is just to consummate some metric properties on Luroth expansion. simultaneously,cites John Wallis、Lagrange、Gauss’s achievements on generalized continued fraction in the late17th century to the18th century.The second chapter mainly gives the basic knowledge of fractal:Hausdorff dimension、measure and Lebesgue measure、Luroth expansions on the real number field、generalized continued fraction and some related basic nature.The third chapter mainly use the knowledge of hausdorff dimension and some related nature of Luroth expansions to prove the hausdorff dimension of a given set is1;and solve the limit problems of partial quotient of Luroth expansions.The forth chapter is by defining the random variable to make it meet the law of large numbers, then apply related knowledge to obtain the measure of sequence convergent to1in generalized continued fraction.