An Introduction To α-continued Fraction And Its Properties

A number has many kinds of expressions; however, comparing with far more widely established systematic fractions, continued fractions, also as an expression, have many advantages. To weigh which expression is the better, a necessary criterion is how clearly the expression reflects this number and whether we can deduce the properties of this number from the expression easily. With regard to above criterion, the preference must be conceded to continued fractions as opposed to systematic fractions. As we know, regular continued fractions and nearest continued fractions have been researched intensely. In this paper, we mainly talk about a similar but different kind of continued fractions---α?continued fractions, which is the extension of regular continued fractions and nearest integer continued fractions. We can get the above mentioned regular continued fractions and nearest integer continued fractions whenα= 1andα= 0.5. Regular continued fractions with much longer research history and much more beautiful results thanα?continued fractions put forward by Nakada in 1981 has form a complete set of theory system. In this paper we mainly talk about properties similar to regular continued fractions and researched by Nakada and so on,the relations betweenα?continued fractions and regular continued fractions and also take the question whetherα?continued fractions have other similar properties into account. In this paper, we have put forward the question about Hausdorff dimension of the set of numbers with bounded partial quotients and got an equivalent conclusion according to properties ofα?continued fractions and the similar result in the case of regular continued fractions.