Distribution Of The Denominator Sequence Modulo1Related To Convergent Factors Of Continued Fractions

Diophantine approximation is a very important branch in studying of number theory.Because of the close connection between continued fraction expansion and diophantineproperty of a real number, the study on continued fractions becomes an important tool instudying Diophantine approximation. Since the terminology”fractal” was introduced for-mally by Mandelbrot in1970s, fractal geometry had been attracted great attention amongthe scientific community. The dimension theory is considered of a fundamental notion infractal Geometry, especially as the most ancient and may also be the most important of theHausdorff dimension based on caratheodory construct has the advantages of the definition inany set, and it is relatively easy to deal with the concept of measure. The set of the Hausdorffdimension provides a degree tool,so it is gaining more and more attentions.This paper is concentrated on studying on the distribution of a sequence modulo1related to convergents of continued fractions. The paper is divided into four parts. We givea brief overview of related backgrounds and current situations in the introductory chapter.In Chapter2, we give cite some definitions, elementary properties and relevant conclusionsfor later use. The last two sections constitute the main part of this thesis. There we studythe distribution of {qnx}n≥1in the interval [0,1], where {qn}n≥1is the sequence of thedenominators of the convergents in the continued fraction expansion of α. Denote(?)By constructing a Moran type subset, the Hausdorff dimension of S(α) is given. In the lastpart, as a corollary, the Hausdorff dimension of the set S(α) can take any value among theunit interval [0,1] as α∈[0,1) varies. That is(?).