Some Problems Concerning The Continued Fraction And Lüroth Expanisons

Due to the comprehensively practical applications of fractal geometry, how to calculate the dimension of fractal set becomes a popular topic. This thesis investigates the Hausdorff dimension of some exceptional sets of continued fraction and Luroth expansions. The primary contributions in this thesis are itemized as follows·In [32], Fernandez, Melian and Pestana study the Hausdorff dimension of the recurrence set of Gauss transformation. Based on the study, Chapter 3 determines the Hausdorff dimension of the recurrence set of Gauss transformation under some conditions. Similarly, in Chapter 6, we consider the Hausdorff dimension of recurrence set on the Laurent series.·Inspired by the above study, Chapter 5 gives the Hausdorff dimension of the recurrence set of Luroth transformation. Under some conditions, we get the Hausdorff dimension of the recurrence set of Liiroth transformation by the mass distribution principle.·In [35], Fiala and Kleban give the definition of sum-level set of continued fraction expansions, and conjecture that the Lebesgue measure of sum-level sets tends to zero. This conjecture is solved by Kessebohmer and Stratmann [61], their proof is mildly spiced with infinite ergodic theory. In Chapter 4, we consider the asymptotic behavior of the Lebesgue measure of the sum-level sets of Luroth expansion, we prove that the Lebesgue measure of these sets vanishes as the level tends to infinity.·In Chapter 7, we study the Hausdorff dimension of a kind of exceptional sets of continued fraction expansion. Under some conditions, we get the Hausdorff dimension of the concerning sets. In the last part, we consider the behavior of the Haar measure of the sum-level sets of continued fraction expansion on the Laurent series field.