Some Problems Concerning The Continued Fraction And Lǖroth Expansions

Due to the comprehensively practical applications of fractal geometry, how tocalculate the dimension of fractal set becomes a popular topic. This thesis investigatesthe Hausdorff dimension of some exceptional sets of continued fraction and Lürothexpansions. The primary contributions in this thesis are itemized as follows? In [32], Ferna′ndez, Meli′an and Pestana study the Hausdorff dimension of therecurrence set of Gauss transformation. Based on the study, Chapter 3 determinesthe Hausdorff dimension of the recurrence set of Gauss transformation undersome conditions. Similarly, in Chapter 6, we consider the Hausdorff dimensionof recurrence set on the Laurent series.? Inspired by the above study,Chapter 5 gives the Hausdorff dimension of therecurrence set of Lüroth transformation. Under some conditions, we get theHausdorff dimension of the recurrence set of Lüroth transformation by the massdistribution principle.? In [35], Fiala and Kleban give the de?nition of sum-level set of continued fractionexpansions, and conjecture that the Lebesgue measure of sum-level sets tendsto zero. This conjecture is solved by Kesseb¨ohmer and Stratmann [61], theirproof is mildly spiced with in?nite ergodic theory. In Chapter 4, we considerthe asymptotic behavior of the Lebesgue measure of the sum-level sets of Lürothexpansion, we prove that the Lebesgue measure of these sets vanishes as the leveltends to in?nity.? In Chapter 7, we study the Hausdorff dimension of a kind of exceptional setsof continued fraction expansion. Under some conditions, we get the Hausdorffdimension of the concerning sets. In the last part, we consider the behavior ofthe Haar measure of the sum-level sets of continued fraction expansion on theLaurent series ?eld.