Theory Of Representations Of Numbers And Fractals

In this thesis,some metric propetries and Hausdorff dimensions of exceptional sets on the real numbers and Laurent series are studied. Including the first chapter of introduction and the second chapter of preliminary,this thesis consists of six chapters.The main content are itemized as following.1.Similar to the real case,for the continued fractions over the Laurent series,in chapter 3, we studied the the iterated logarithm for the largest degree of the quotients,and based on which, the exceptional set (?)(α≥0)was considered.Applying the Lipschitz invatiance of Hausdorff dimension to the suitable constructed subset of E(α),we obtained the Hausdorff dimension of E(α).2.For Luroth expansions,we concerned with the Hausdorff dimension of the sets determined by the digits obeying some restrictions. In chapter 4,similar to the results of Luczak,we obtained the Hausdorff dimensions of the following sets:E(a,b)={x∈(0,1]:dn(x)≥(?) (?)3.In chapter 5,we studied some exceptional set of Engel expansions.In 1973,J.Galam-bos considered the metric properties of the growth ratio of the Oppenheim expansions,and posed the problem on studying the size of the set{x∈(0,1]:1≤Tn(x)≤m}.For the Engel case,by suitable constructing a mass distribution,We obtained the Hausdorff dimension of the set (?)4.For the general continued fractions(GCF)expansions posed by Schweiger,under the restriction of-1<(?)≤1,we obtained some important metric properties,such as"0-1"law, central limit theorem and iterated logarithm laws.