On Problems Of Continued Fractions And Diophantine Approximation Over The Field Of Formal Laurent Series

In this dissertation, we are concerned with some problems of continued fractions and Diophantine approximation over the field of formal Laurent series. We get the Hausdorff dimensions of exceptional sets of continued fractions whose partial quotients satisfy some restricted conditions, and we prove a result about Diophantine approximation over the field of formal Laurent series. Including the first chapter of introduction and the second chapter of preliminary, there are five chapters in the thesis.It is well known that, with respect to the Haar measure, almost surely, the sum of degrees of partial quotients grows linearly. In 2005, J. Wu studied the cases with polynomial and exponential growth rate, and gave the Hausdorff dimensions of these exceptional sets. In chapter three, we quantify the exceptional sets of points with a generally functional order faster than linear order ones by their Hausdorff dimension, which covers the earlier results by J. Wu.In 2010, M. Jellali et al introduced a new kind of continued fractions algorithm over the field of formal Laurent series, calledβ-continued fractions. The metric and ergodic properties of this new dynamical system have been studied by the authors. Furthermore, they considered bounded-type set and set of Laurent series having a given rate of convergence, and gave their Hausdorff dimensions. In chapter four, we first discuss metric properties of the digits｛en(Χ), n≧ 1} occurring inβ-continued fractions, obtain the so-called "0-1" law and limit results on the digits. Then, we studyβ-continued fractions with sequences of partial quotients andβ-continued fractions whose sum of degrees of partial quotients tends to infinity with generally functional growth rate, we give their Hausdorff dimensions.For any functionψ:R≥1→R≥0. Let Let b≥3 be an integer and J(b) be a proper subset of{0,1,···, b-1} with at least two elements. Denote KJ(b) the set of real numbers in [0,1] whose b-ary expansion consists exclusively of digits in J(b). Suppose the function x2ψ(x) is non-increasing and tends to 0, Y. Bugeaud showed that:For any positive real number c less than 1/b, the set (K(ψ)\K(cψ)))∩KJ(b) is uncountable. In chapter five, we consider the analogous problems over the field of formal Laurent series, and get a result about Diophantine approximation onβ-expansion.