Representation Of Numbers And Fractals

In this dissertation, we are concerned with some problems in continued fraction, dyadicexpansion and generalized random moran sets.Including the first chapter of introduction, thereare five chapters in the thesis.The main results in this thesis are in the chapter 3, chapter 4 andchapter 5.In the chapter 3, we study the size of the set consisting of all Szmer(?)di points.In thechapter 4, we study the Hausdorff dimension of the maximal run-length in dyadic expansion.Andin the chapter 5, we study the packing dimension of generalized random Moran sets.Specificcases are in the following:1.How many points contain arithmetic progressions in continued fraction expansion?For any irrational x∈[0, 1), let [a₁(x),a₂(x),…] be its continued fraction expansion.Szemer(?)dishowed that an integer subset contains arbitrarily long arithmetic progressions, if itis of positive density in N.Inspired by Szemer(?)di's theorem, we are led to ask the followingquestions naturally:(1).Which point is a Szemer(?)di point in continued fraction expansion?(2).How large the set of such points is?The first question is answered by the definition of Szmer(?)di points in this note.With respectto the latter, we'll quantify the Hausdorff dimension of the set consisting of all Szmer(?)di pointsby principle of the mass distribution.2.On the Hausdorff dimension of the maximal run-length in dyadic expansion.Let x = (ε₁,ε₂,…, ) be its dyadic expansion of x∈[0, 1).Call r_n(x):= max{j:ε_i+1=…=ε_i+j=1,0≤i≤n-j} the n-th maximal run-length of x.R(?)nyi showed that, for almostall x∈[0, 1), (?) r_n(x)/log n=1.In this paper, the sets of different asymptotic properties on r_n arestudied.Moreover, We obtain the Hausdorff dimensions of the following sets by principle of themass distribution,where {δ_n}_n=1^∞is a non-decreasing integer sequence satisfyingδ_n→∞with n→∞. 3.Packing dimension of generalized random Moran setsIn the 1980s, Falconer, Graf, Mauldin and Williams investigated random fractal sets by randomizingeach step in Moran's deterministic constructions.They studied the geometric propertiesof such random constructions and obtained the Hausdorff dimensions of the random fractals associatedwith such random constructions.Liu, Wen and Wu generalized the works of Falconer,Graf, Mauldin and Williams to the case that at each stage, the contracting vectors are not identicallydistributed.In this paper, we get the packing dimensions of the generalized random Moransets in the Liu, Wen and Wu.The original packing dimension is defined by balls packing, which is more difficult thanby basic intervals in our case, therefor, we define a new dimension equivalent with it so that wecan pack or cover the sets by basic intervals.Moreover, Liu obtain the Hausdorff dimensions byproving the existence of the moments of all order and giving a common bound of these moments.However, it is not so obvious and natural in the case of calculating the packing dimensions.Weovercome this difficulty to prove our conclusion with estimating the negative moments.