An Application of Fibonacci Sequence on Continued Fractions

Transcription

1 International Mathematical Forum, Vol. 0, 205, no. 2, HIKARI Ltd, An Application of Fibonacci Sequence on Continued Fractions Ali H. Hakami Department of Mathematics, Faculty of Science, Jazan University P.O. Box 277, Jazan, Postal Code: 4542, Saudi Arabia Copyright c 205 Ali H. Hakami. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract Let F 0 = 0, F =, F 2 =,... be the Fibonacci sequence. Fix k N. We prove that for almost every x 0, ), the pattern,,..., k-digits) appears in the continued fraction expansion x = [a, a 2,...] with frequency ) k log + ) k F 2 k+ )/. Mathematics Subject Classification: Primary B39, A55 Keywords: Fibonacci sequence, Fibonacci numbers, continue fractions Introduction The F ibonacci sequence F n ) is defined by F 0 = 0, F =, and for n 2, F n = F n + F n 2. The reader will find an introduction to this well-studied sequence in the books by Koshy [8] and Moll [0]. An expression of the form a 0 + a + a 2 + a 3 +

2 70 Ali H. Hakami where a 0, a, a 2,... R and a, a 2,... > 0 is said to be infinite continued fraction and is denoted concisely by [a 0 ; a, a 2,...]. An infinite continued fraction is said to be simple if a 0, a, a 2,... Z. For more detail about of the basic properties of continued fractions and its related by Fibonacci sequence can be found in Jones and Thron [5], Khovanskii [6], Khinchin [7] and Lorentzen and Waadeland [9]. In this paper we are seeking to prove Theorem. Let F 0 = 0, F =, F 2 =,... be the Fibonacci sequence. Fix k N. Then for almost every x 0, ), the pattern,,..., k-digits) appears in the continued fraction expansion x = [a, a 2,...] with frequency ) k log + ) k F 2 k+ )/, that is: lim n n # {j {,..., n} : a j = a j+ = = a j+k = } = ) k log ) + ) k F 2 k+2. We shall devote section 3 to give the proof of Theorem. Throughout this work, we assume that the reader has familiarity with some introductory number theory suggested references see books by Borevich and Shafarevich [], Hardy and Wright[4], Niven, Zuckerman and Montogomery[], and Stark[4]), ergodic theory see for example book by Einsiedler and Ward [2]), analysis and measure theory see books by Rudin[2,3] and G.B. Folland [3]). Simple facts about concepts of Gauss measure, Gauss map also are needed see for example ref. Einsiedler and Ward [2]). 2 Basic Lemmas To prove Theorem. we shall need the following lemma Lemma 2. Let k N. For x Y = 0, )\Q let a x), a 2 x),... be the digits of its continued fraction expansion. Let I k 0, ) be the interval I k = F k, ) if k is even, and I k = F, k ) if k is odd. Then a x) = a 2 x) = = a k x) = holds if and only if x I k. The next two lemmas help us to prove Lemma 2. and Theorem.. Lemma 2.2 [2], Theorem 2.30) Let X, B, µ, T ) be a measure-preserving system. If f L µ, then n lim ft j x) = f x) n n j=0

3 Fibonacci sequence and continued fractions 7 converges almost everywhere and in L µ to a T -invariant function f L µ, and f dµ = f dµ. If T is ergodic, then almost everywhere. f x) = f dµ Lemma 2.3 [2], Theorem 3.7) The continued fraction map T x) = { x } on 0, ) is ergodic with respect to the Gauss measure µ. Proof of Lemma 2.. We have a = x. Hence a = if and only if x >, 2 i.e. if and only if x, ) = I 2, and the lemma is proved in the case k =. Now assume that the lemma holds for a given k N. Recall that the Gauss map x {x } on Y has the effect of shifting the continued fraction of x one step to the left see Einsiedler and Ward [2], p. 79). Hence we have a = a 2 = = a k+ = if and only if a = and {x } I k, viz., if and only if x, ) and 2 {x } = x I k. But I k is the open interval with F endpoints k and ; also + F k ) = F k + = and similarly + ) = F k+3 ; hence x I k holds if and only if x lies in the open interval with endpoints and F k+3. The computation just carried out F k > also shows that < F k+3 if and vice versa; hence the open interval just referred to is in fact I k+, i.e. we have proved that x I k holds if and only if x I k+. Note also that I k+, ), since for all k we 2 have F k and = F k + F k 2F k. We have thus proved that a = a 2 = = a k+ = holds if and only if x I k+, i.e. the lemma holds also for k +. Hence by induction, the lemma holds for all k N. 3 Proof of Theorem. Let Y = 0, )\Q as above, let µ be the Gauss measure i.e. dµx) = dx +x ) and let T : Y Y be the Gauss map. Using Lemma 2.3, T is ergodic. Now, for fixed k N, let f : Y R be the characterstic function of the interval I k. Clearly f L Y, µ); hence the pointwise Ergodic Theorem Lemma 2.2) applies, and we conclude that for µ-almost all x Y we have n lim ft j x) = f dµ. 3.) n n j= Y

4 72 Ali H. Hakami But here, by Lemma 2. and using the fact that T corresponds to left shifting the continued fraction expansion, we have ft j x) = if and only if a j x) = a j+ x) = = a j+k x) =, and therefore the left hand side of 3.) equals lim n n #{j {,..., n} : a jx) = a j+ x) = = a j+k x) = }. 3.2) On the other hand, the right hand side is: Y f dµ = µi k ) = = )k = )k I k dx + x log log Fk+ ) )) Fk + log + ) )) Fk+2 log Fk+3 ) = )k log Fk+3 Fk+2 2 log + ) k F 2 = ) k k+2 ) 3.3) where in the last step we used the formula F k+3 = F 2 k+2 + ) k. To see this formula, since F k+3 = +, the statement is equivalent with + F 2 k+ F 2 k+2 = )k. Note that this identity holds for k = 0, and note also that + F 2 k+ F 2 k+2 = ) + F 2 k+ = F k + ) F k ) + F 2 k+ = F k + F 2 k F 2 k+). Hence the claim follows by induction. We have thus proved that for µ-almost all x Y equivalently, for Lebesgue almost all x Y ),) the limit in 3.2) equals the expression in 3.3). Acknowledgments The author is grateful to Jazan University for providing excellent research facilities.

5 Fibonacci sequence and continued fractions 73 References [] Z. I. Borevich and I.R. Shafarevich, Number Theory, Vol. 20 in Series on Pure and Applied Mathematics, New York, 966. [2] M. Einsiedler and T. Ward, Ergodic Theory with a view towards Number Theory, Springer Graduate Text in Mathematics, Vol. 259, Springer- Verlag London Ltd., London, [3] G. B. Folland, Real Analysis: Modern Techniques and Their Applications, 2nd edn, John Wiley and Sons, 999. [4] G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Oxford Science Publications, Clarenden Press, Oxford, 998 [5] W. B. Jones and W. J. Thron, Continued Fractions. Analytic Theory and Applications, Vol. of Encyclopedia of Mathematics and its Applications. Addison-Wesley Publishing Co., Reading, Mass., 980. [6] A. N. Khovanskii, The application of continued fractions and their generalizations to problems in approximation theory. Translated by P. Wynn. P. Noordhoff N. V., Groningen, 963. [7] A. Y. Khinchin, Continued fractions. With a preface by B. V. Gnedenko. Translated from the third 96) Russian edition. Reprint of the 964 translation. Dover Publications, Inc., Mineola, Ny, 997. [8] T. Koshy, Fibonacci and Lucas numbers with applications. Pure and Applied Mathematics, New York). Wiley-Interscience, New York, 200. [9] L. Lorentzen and H. Waadeland, Continued fractions with applications, Vol. 3 of Studies in Computational Mathematics. North-Holland Publishing Co., Amsterdam, 992. [0] V. H. Moll, Number and functions, Student Mathematical Library. American Mathematical Society, Providence, RI, 202. Special Functions for Undergraduates. [] I. Niven, H. S. Zuckerman and H. L. Montogomery, An introduction to the Theory of Numbers, John Wiley and Sons, New Yourk, 99. [2] W. Rudin, Principles of Mathematical Analysis, 3nd edn, McGraw-Hill, 976.

6 74 Ali H. Hakami [3] W. Rudin, Real and Complex Analysis, 3nd edn, McGraw-Hill, 987. [4] H. M. Stark, An Introduction to Number Theory, Markham Publishing Company, Chicago, 970. Received: January 3, 205; Published: January 25, 205