Abstract :
For any irrationalxset membership, variant[0, 1] we denote bypn(x)/qn(x),n=1, 2, … the sequence of its continued fraction convergents and defineθn(x) colon, equalsqn qnx−pn. Also letT: [0, 1]→[0, 1] be defined byT(0)=0 andT(x)=1/x−[1/x] ifx≠0. For some random variablesX1, X2, …, which are connected with the regular continued fraction expansion, the subadditive ergodic theorem yields to the existence of a functionωsatisfying: for allzset membership, variantimage,imageIn particular, forXn=θn, using this study and a result of Knuth, we give another proof of the following conjecture of Lenstra (the first proof of this conjecture has been given by Bosma, Jager, and Wiedijk): for allzset membership, variant[0, 1],[formula]for almost everyx. Furthermore, forXn=θnring operatorTnandXn=(qn−1/qn)ring operatorTn, the functionsωare explicitly determined. The above results show that the subadditive ergodic theorem can be useful in the metric theory of continued fraction.
