The Fractional Dimension Of Continued Fractions With Partial Quotients

The fractional dimensional theory of continued fractions was introduced by Janik in his former work. He consider the set E of continued fractions whose partial quotients do not exceedαand the set Ea whose partial quotients do not exceedα. He proves that the Hausdorff dimension of E is one and E2 is between 1/4 and 1.Later on,I. J.Good obtained the dimension of Eαin terms of the Euler polynominals used in the theory of continued fractions.By this means the fractional dimension of Eαcould be calculated to any desired degree of accuracy,and in particular he proved that the Hausdorff dimension of E2 is between 0.5306 and 0.5320.In 1941, I. J Good investigated the fractional dimension of sets of contin-ued fractions whose partial quotients {αn} obey various conditions. Good proved that the set of continued fractions whose partial quotients tend to infinity has Hausdorff dimension 1/2. In 1967, Hirst make a further research of this result whereαn tends to infinity rapidly. In all these results the only restrictions on {αn} are of the typeαn≥f(n) and {αn} tend to infinity. In this paper, we discussed the main nature of the fractional dimension of sets of continued fractions and proved analogous results concerning the cases where {αn} is further restricted to some sequences of natural numbers and we discuss the Hausdorff dimension of continued fraction's partial quotients in certain conditions.This paper is divided into three parts, the main content specific as fol-lows:The first part mainly introduced the background and the development of the fractional dimension, and point out its specific meaning. Finally, We presents the main content of this article and relevant results. The second part gives the knowledge of the Hausdorff dimension of related definitions and properties, The third part is divided into two section, the first quar-ter introduced the fractional dimension of relevant properties, the second quarter gives the fractional dimension's estimation problem When even the partial quotients are mainly discussed in positive integer and give a sequence of relevant conclusions. Finally we introduces the method using the reduced order to estimate fractioal dimension of continued fractions.