The Distribution Pattern And Properties Of Quadratic Intervals

In this paper, we briefly give an introduction of the theory of continued fraction and string of integer which are the main tool of our analysis. Our discussion focuses on the continued fraction expansion of real numbers lying in ]1,0(, especially the rationals which have two presentations. In addition, we also discuss the relation of different convergents and the link between the value and the parity of the serial number of partial quotient's position. Then naturally we introduce the knowledge of strings about which we emphasize the determination of value of strings with equal length. After all, we construct a family of open intervals contained in(0, 1], the union of which is noted as M. Since the endpoints of every interval are both quadratic surds, we call them quadratic interval.About this family, we put our emphasis on the connection between interval's position and its pseudocenter, and then we can determine the relationshiop of two quadratic intervals by the feature of the pseudocenter. The introduction of maximal interval makes it easier for us to learn the distribute pattern of these intervals. On strict theoretical base, we construct a series of maximal intervals, and note their union as F. It's easy to have the following conclusion: FM ?.In this paper, we also introduce the determination of maximal intervals and the complement of F, namely ??M\]1,0(. At last, we analyse the feature of its elements,prove this set is uncountable and figure out that its Hausdorff dimension equals 1.