A Note on the Approximation by Continued Fractions under an Extra Condition

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. In this note the distribution of the approximation coefficients \\Theta n , associated with the regular continued fraction expansion of numbers x 2 [0; 1), is given under extra conditions on the numerators and denominators of the convergents pn=qn . Similar results are also obtained for S-expansions. Further, a Gauss-Kusmin type theorem is derived for the regular continued fraction expansion under these extra conditions. Contents 1. Introduction 69 2. A Natural Extension of a Skew Product by Jager and Liardet 70 3. S-expansions 76 References 79 1. Introduction A classical result by Hurwitz states that for every irrational number x there exist infinitely many pairs of (co-prime) integers p and q, q ? 0, such that fi fi fi fi x \\Gamma p q fi fi fi fi ! 1 p 5 1 q 2 : (1) In the past century a great number of papers appeared, aimed at reproving, refining or generalizing Hurwitz' result (1). Here we mention a theorem by Koksma [Kok], which in itself was a refinement of a resul..

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