On The Growth Of The Denominators Of Convergents

In this thesis, we are concerned with the growth of the denominators of the convergents and the Levy constant for quadratic irrationals in continued fractions.As we know, the continued fractions play an important role in the investigation of irrational numbers. To every real number x there is corresponds, uniquely, a continued fraction whose value is this number. This continued fraction terminates if x is rational. If x is irrational, it is infinite. We denote the canonical representation of x by (( )),and called it the convergent of order n of x . As it is well-known we have for all n≥0 the diophintine inequality which proves that the sequence (( ))convergents to x since ( )The inequality (1) shows that the (( ))q x will be very good approximations of x if ( )q nx increase quickly.The"general"behaviour of qn when n→∞is given by a famous theorem of P. Levy which asserts that for almost all irrational numbers x (in the sense of Lebesgue),its Levy constant exists and equals toπ212log 2. That is to say, the set of numbers whose Levy constant does not exsist or it exist but not equal toπ212log 2 is of Lebesgue measure zero. Faivre had proved that the Levy constant for quadratic irrationals exsisted and employed the ergodic theorem to prove that the set of Levy constant for the numbers in the zero-Lebesgue measure set is dense in .Furthermore, Baxa proved that there existed non-denumerably many pairwise not equivalent irrational numbers in the zero-Lebesgue measure set. J. Wu improve this result, he obtained the lower bound of the hausdorff dimension of the zero-Lebesgue measure set. On the basis of these, this paper use a more simple method to prove the remark in Baxa's paper, and use the method in J. Wu's paper which is used to obtain the lower bound of the hausdorff dimension of the zero-Lebesgue measure set to prove that the set of levy constant for quadratic irrationals is dense in .