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

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

New York Journal of MathematicsNew York J. Math. 3A (1998) 69{80.A Note on the Approximation by ContinuedFractions under an Extra ConditionKarma Dajani and Cor KraaikampAbstract. In this note the distribution of the approximation coecients 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 theconvergents pn=qn. Similar results are also obtained for S-expansions. Further,a Gauss-Kusmin type theorem is derived for the regular continued fractionexpansion under these extra conditions.Contents1. Introduction 692. A Natural Extension of a Skew Product by Jager and Liardet 703. S-expansions 76References 791. IntroductionA classical result by Hurwitz states that for every irrational number x there existinnitely many pairs of (co-prime) integers p and q, q 0, such thatx pq 1p5 1q2 :(1)In the past century a great number of papers appeared, aimed at reproving, reningor generalizing Hurwitz result (1). Here we mention a theorem by Koksma [Kok],which in itself was a renement of a result by Hartman [H].Theorem 1. (Koksma) Let x be an irrational real number, and let m 1; a andb be integers. Then for every 0 there exist innitely many pairs of integers pand q, q 0, such thatx pq m2(1 + )p5 1q2 and p a(mod m); q b(mod m):(2)The constant p5 is best possible, i.e., it can not be replaced by a larger one.Received November 24, 1997.Mathematics Subject Classication. 28D05, 11K50.Key words and phrases. continued fractions, natural extension, skew-product.c 1998 State University of New YorkISSN 1076-9803/9869 70 Karma Dajani and Cor KraaikampThe proof of this theorem, and also that of related results by Descombes andPoitou [DP], rests on the strong approximation properties of the regular continuedfraction.In this note we will determine for almost all1 x the asymtotic density of thoseregular continued fraction convergents pn=qn of x satisfying (2). To b