Sets Of Exact Diophantine Approximation, Oppenheim Expansion And Some Related Problems

In this dissertation, we are concerned with set of exact Diophantine approximation over the field of formal Laurcnl series. We discuss the problem whether set of exact Diophantine approx-imation is a null set, and give its dimension results when the error function in non-increasing. Moreover, we study the efficiency of approximating real numbers for Luroth expansion and some exceptional sets related to the Oppenheim continued fraction expansion. Including the first chapter of introduction and the second chapter of preliminary, there are five chapters in the thesis.Given a functionψ:R＞0→R＞0 with tψ(x)= o(x-2), let Exact(ψ) be the set of exact Diophantine approximation, namely the set of real numbers that are approximable by rational numbers to orderψ;, but to no order cψwith 0＜c＜1. It is unknown whether the set Exact.(ψ) is empty, but whenψis non-increasing the Hausdorff dimension of the set Eract(ψ) is known to be 2/λ, whereλis the lower order at infinity of the function 1/ψ. In third chapter, over the field of Laurent series we prove the set of exact Diophantine approximation is uncountable, and when the error function is non-increasing we give its Hausdorff dimension analogous to the real case. Furthermore, we give a metric result about the set of Laurent series that are approximable to exact order.The convergents in the continued fraction expansion of real number x are all optimal ap-proximation to x, also, x can be well approximated by its convergents in the continued fraction expansion. However, the situation is very different in Liiroth expansion. In fourth chapter, we show that the set of real numbers which can have convergents as the optimal approximation for infinitely many times in Luroth expansion is null, this implies that almost no points can be well approximated by the convergents in Liiroth expansion. Consequently, Hausdorff dimension is introduced to quantify these exceptional sets.It is well-known to us that regular continued fraction expansion, Engel continued fraction expansion and Sylvester continued fraction expansion are included in Oppenheim continued fraction expansion. In fifth chapter, we study the exceptional sets determined by partial quotients in its Engel and Sylvester continued fraction expansion respectively, the set consisting of the points whose speed of convergence is faster than the "typical" points for Engel continued fraction expansion, moreover, we study the set of points with the same Engel and Sylvester continued fraction expansion.