Metric Properties For The Oppenheim Expansion Over The Field Of Laurent Series

Metric number theory is an important branch in number theory. For any real number, it can be represented by more than20kinds of forms presently. Its metric properties and fractal structure help us to describe and realize real number from different angels. In this thesis, an iterated logarithm law and the corresponding exceptional set of the Oppenheim expansion over the formal series are studied,which enriches the metric theory and the dimension theory, also, it supplies the theoretical principle for the fractal applications.In chapter1, the purpose, significance, the current situation and the application of that we researching are introduced.In chapter2, we showed the definition of the fractal and several fractal dimension at the beginning, focuses on the development of the Hausdorff dimension. And then we have introduced the definition of Oppenheim expansion over the formal series and the some important metric properties.In chapter3, based on the Borel-Cantelli lemma and the proof method which J.Galambos proved the iterated logarithm law, we proved that for {â–³n(x)}, the iterated logarithm law holds, which is the foundation of the exceptional set study.In chapter4, for the Oppenheim expansion over the formal series, by applying the Lipschitz properties of Hausdorff dimension to the suitable chosen Cantor set, we obtained the Hausdorff dimension of the set is full.