Metrical Theory of Continued Fractions

M. Iosifescu

• 30 september 2002
• 9781402008924

Samenvatting:

The positive integers an(w) = al(T - (W)), n E N+ = {1,2··· }, where al(w) = integer part of 1/w, w E 0, are called the (regular continued fraction) digits of w. n, we have w = lim [al(w),··· , an(w)], w E 0, n--->oo thus explaining the name of T.



This is a complete treatment of the metrical theory of regular continued fractions and related representations of real numbers. The authors have attempted to give the best possible results now known, with proofs that are the simplest and most direct. In order to unify and generalize the results obtained so far, additional concepts have been introduced, for example: an infinite order chain representation of the continued fraction expansion of irrationals and the conditional measures associated with, and the extended random variables corresponding to, that representation. Also, procedures as singularization and insertion allow us to obtain most of the continued fraction expansions related to the regular continued fraction expansion. The authors present and prove with full details for the first time in book form, the most recent developments in solving the celebrated 1812 Gauss' problem with originated the metrical theory of continued fractions. At the same time, they study exhaustively the Perron-Frobenius operator, which is of basic importance in this theory, on various Banach spaces including that of functions of bounded variation on the unit interval.