FAREY-PELL SEQUENCE, APPROXIMATION TO IRRATIONALS AND HURWITZ S INEQUALITY (COMMUNICATED BY TOUFIK MANSOUR)

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1 Bulletin of Mathematical Analysis Applications ISSN: 8-9 URL: Volume 8 Issue (06) Pages - FAREY-PELL SEQUENCE APPROXIMATION TO IRRATIONALS AND HURWITZ S INEQUALITY (COMMUNICATED BY TOUFIK MANSOUR) ILKER AKKUS NURETTIN IRMAK AND GONCA KIZILASLAN Astract The purpose of this paper is to give the notion of Farey-Pell sequence We investigate some identities of the Farey-Pell sequence Finally a generalization of Farey-Pell sequence an approximation to irrationals via Farey-Pell fractions are given Introduction A Farey sequence of der n is the sequence of fractions p q where p q are coprimes 0 p < q n arranged in der of increasing size This sequence has a rich histy properties Let a c d e two consecutive elements of this sequence A well-known identity of the Farey sequence is c ad = which is known as neigh identity Alladi [] defined the concept of Farey sequence of Fionacci numers In this study Alladi investigate some relations etween Farey Fionacci fractions Then Matyas [4] generalized the idea of Farey Fionacci sequence gave the sufficient conditions f the sequence G k = AG k +BG k so that the properties of points of symmetry hold in this me general setting Alladi [3] gave the est approximation to irrational numers y Farey Fionacci fractions In this paper we go a step further We define Farey-Pell sequence using Pell numers Although it seems that the identities in Farey Fionacci sequence have to hold we need to define the concept of center of interval since Pell sequence does not hold the same recurrence with Fionacci sequence Afterwards we give the approximation to irrational numers via Farey-Pell fractions Thus it will e convenient to give the definition of Pell sequence F n the Pell sequence { } is defined y the following recurrence relation = + with the initial conditions P 0 = 0 P = A few Pell numers are Its Binet fmula is known as = αn β n α β 000 Mathematics Suject Classification B57 4A0 Key wds phrases Farey sequence; Farey-Pell sequence; approximation y rational functions c 06 Universiteti i Prishtinës Prishtinë Kosovë Sumitted January 6 06 Pulished Feruary 5 06

2 ILKER AKKUS NURETTIN IRMAK AND GONCA KIZILASLAN where α β are the roots of the characteristic equation x x = 0 Farey-Pell Sequence its identities Definition The Farey-Pell sequence of der is the set of all possile fractions Pi 0 arranged in ascending der of size 0/ is the first fraction of the sequence This sequence is denoted y F Further we denote the r th fraction of the F y FP (r)n We give a Farey-Pell sequence of der 9 as follows Definition We define the fraction FP (r)n as a point of symmetry if FP (r )n FP (r+)n have the same denominat Definition 3 An Fnterval contains the set of all F fractions etween two consecutive points of symmetry Definition 4 We define the center of an Fnterval as a fraction that has the greatest denominat of all fractions in the interval Definition 5 [] We define the distance etween FP (r)k FP (k)n as r k Definition 6 [] Weighted mediant of two fractions a c d a + c + d a + c + d is defined as Theem 7 FP (r+k+)n FP (r k)n have the same denominat if only if one of them coincides with a point of symmetry Proof In the F sequence the terms are arranged in the following sense The terms in the last Fnterval are of the fm Pj The terms in the Fnterval previous to the last Fnterval are of the fm Pj The weighted mediant of two fractions Pi Pi which is Pi lies in etween them That is If < then < < If < then < < We use induction as the method of proof It is true f all F sequence up to 69 Let us consider 69 as F the F sequence of der the following fractions need to e placed: P P 3 will e exactly etween Pi Pi Assume that Pi < Pi < Pi The from the point of symmetry say is one me than the distance distance of Pi from that point of symmetry Hence it is true f 408 Continuing in this way it can e shown f

3 APPROXIMATION TO IRRATIONALS VIA FAREY-PELL FRACTIONS 3 Theem 8 F an Fnterval [ ] the denominat of the fraction next to is the denominat of the next fraction is + then +4 till we reach the center of the F sequence ie until +k does not exceed Then the denominat of the fraction after +k will e the maximum possile fraction not greater than not equal to any of the terms created Thus it is either +k+ +k say After the denominat of the fractions will e 4 till we reach Proof By induction on Theem 7 gives the result h Theem 9 h k h are three consecutive fractions in FP n such that neither of them is center of any Fnterval point of symmetry Then the following holds: = h + h k + h Proof If h k h are three consecutive fractions in any Fnterval which are not center of Fnterval point of symmetry then there are two possiilities Case : Let h k = Pi h = Pi h = Pi+ + In this case it is ovious that Case : Similarly if h k = Pi+ + h + h k + = Pi h = 6 6 = = Pi then we have = h + h k + When one of the three consecutive fractions is the center of an Fnterval we have the following result Theem 0 If one of the three consecutive fractions h k of any interval then we have the followings: () If h () If h k is the center of Fnterval then we have either = h + h + k is the center of Fnterval then we have = h + h + k h is the center = h + h + k = h + h + k (3) If h is the center of Fnterval then we have = h + h + k = h + h + k

4 4 ILKER AKKUS NURETTIN IRMAK AND GONCA KIZILASLAN Proof We give the proof of first item Let h k e the center of any Fnterval Then two cases may e due to the dering of the sequence F Firstly if h k = h = Pi h = Pi+ + then we otain = h + h + k y the recurrence of Pell sequence F the second case that is if h k = Pi h h = Pi+ + we have The other items can e proven similarly Theem Let h k h = h + h + k = Pi e consecutive fractions of an F sequence then = h + k + Proof From Theem 8 it follows that h k = 5 + h = + Thus = = = h k + = + + = (5 + ) = (h + ) k + Now we present a theem which gives a relation aout three consecutive fractions depending upon including center of an Fnterval not Theem Let h k h h that < h k < h sequence: < h < () If h k is the center of an Fnterval say h equal to Pi say h k = Pi () If neither of h k e three consecutive fractions in an Fnterval such f i Then the following holds f an F k h = h = h = Pi then either h k is the center of an Fnterval then k h = h Proof Since the proof depends on the proof of the Theem 0 we omit the proof to cut unnecessary repetition Collary 3 If h k h h that < h k < h are three consecutive fractions in an Fnterval such < h < then we have Proof It is clear from Theem k h () is

5 APPROXIMATION TO IRRATIONALS VIA FAREY-PELL FRACTIONS 5 Lemma 4 If j i = j i > 0 then = P j j j = P j j j Proof We apply Binet s fmula that where = αn β n α β α = + β = Using this fmula the proof can e done similar to [] Lemma Collary 5 [] If Hence Pi are in the same Fnterval then = P j j i = P j j i < is an integral multiple of i i Here i ( i ) is the term otained y the difference in indices of the numerat denominat of each fraction of that Fnterval [ Definition 6 [] F an Fnterval ] two fractions a a this Fnterval if the distance of a from equals the distance of a from then these two fractions are called conjugate fractions in this Fnterval Theem 7 If a a Proof Let a = P k Then since a Thus we otain y Collary 5 [ are conjugate in an Fnterval a a = ] then a are conjugate fractions we have a = P k+ + a a = P k+ + P k = P k = ( Definition 8 [] Let a e a fraction in the Fnterval f a is the dered FP pair [( a ) ( )] a Theem 9 Let [( a ) ( )] a e a couplet f a Then we have a = P k ) in The couplet where P k is the k th Pell numer a = P k+

6 6 ILKER AKKUS NURETTIN IRMAK AND GONCA KIZILASLAN Proof Let a = Pj i+ Then y Collary 5 a is a is i+ = P k i+ = P k+ Multiply aove equation y then adding the equations we otain Thus i+ = P k+ j i = k ie i+ = P k+ i+ = i We can estalish the last equation y Lemma 4 ( Definition 0 [] If a a are conjugate fractions of the Fnterval then [( a ) ( )] [( ) ( )] a a a are said to e conjugate couplets ) Theem Let [( a ) ( )] a e conjugate couplets If [( ) ( )] a a a = P k a = P k+ then a = P k+ a = P k+ Proof In Theem 9 j i is the difference in the indices of If a = Pj i+ then k = j i Since a is conjugate with a a we have = Pj i+3 + Thus in the equation f a a = P m the index of the constant fact P m will e m = j i + = k + That is a = P k+ Therefe a = P k+ y Theem 9 Theem If FP (r)n is a point of symmetry then r { } that is the sequence of distance etween two consecutive points of symmetry will e Proof We have to show that if there are n terms in an Fnterval then there are (n+) terms in the next Let there e k terms of the fm / Clearly there are k + terms of the fm + / these fractions are in next to the Fnterval in which the k terms of the fm / lie Thus the sequence is an arithmetic progression with common difference of Also the second term is always / Hence we have the result Here j i is assumed constant

7 APPROXIMATION TO IRRATIONALS VIA FAREY-PELL FRACTIONS 7 Generalized F Sequence We defined the F sequence in the interval [0 ] We now define it in the interval [0 ) Definition 3 The F sequence of der is the set of all fractions Pi j n arranged in ascending der of magnitude i j 0 If i j then the F sequence is in the interval [0 ] Definition 4 Suppose that FP (r)n > If FP (r )n FP (r+)n have the same numerat then FP (r)n is a point of symmetry Definition 5 A fraction with denominat is called a center Now we give some results f F sequence in the interval [0 ) Their proofs can e done similarly to the proofs of the F sequence in the interval [0 ] Theem 6 If FP (r)n > is a point of symmetry then FP (r+k)n FP (r k )n have the same numerat until one of them coincides with a center Theem 7 A point of symmetry is the fraction which has either numerat denominat Theem 8 F fractions greater than any Fnterval is given y [ ] 3 Approximation to irrationals with Farey-Pell fractions In this section we present the approximation of irrationals with Farey-Pell fractions Definition 3 F any F we fm a new dered set F consisting all rationals in F together with mediants of consecutive rationals in F define Fr+ as all rationals in Fr with their mediants of consecutive rationals We define P = Fr r= Proposition 3 P is dense in the interval [0 ) Thus we can approximate every irrational τ y a sequence of rationals a in P Let s take the irrational τ > 0 F τ < 0 we can approximate y a where a in P Befe giving further we give a lemma which helps us to prove the main theem f this section Lemma 33 There does not exist integers x y such that the inequalities xy ( 5 x + ) y simultaneously hold x (x + y ) ( ) 5 x + x + y F the proof one can see [5] (Lemma 4 pp 5) In the following theems we give a est approximation f some irrational numers If a/ < c/d then (a + c) / ( + d) is the mediant fraction of these fractions

8 8 ILKER AKKUS NURETTIN IRMAK AND GONCA KIZILASLAN Theem 34 Suppose that τ is an irrational numer etween two consecutive fractions a a where one of the fractions is the center of an Fnterval say P m Pn the other one is Pm in the sequence F Then f every integer n there exist infinitely many fractions h k P such that τ h k < (3) 5k Meover 5 is the est possile constant ie if 5 is replaced y a igger constant the assertion fails Proof Assume that τ < let τ is etween two consecutive points of symmetry say etween two consecutive fractions a a where one of the fractions is the center of an Fnterval say Pm the other one is Pm in F F each positive integers n there are successive fractions x y u v in Fr [ ] x so that τ y u v It is clear that x y < x + u y + v < u v x + u y + v Fr+ Then there are two cases namely we have either x y < x + u y + v < τ < u v x y < τ < x + u y + v < u v Assume that x y < x + u y + v < τ < u v none of these fractions satisfy the inequality (3) Then τ x y 5y ; τ x + u y + v ; 5 (y + v) u v τ 5v If we comine these three inequalities we otain u v x y P ( i 5 v + ) y u v x + u y + v P ( ) i 5 v + y + v From the inequality () we have vy ( 5 v + ) y v (y + v) ( ) 5 v + y + v which is a contradiction accding to the Lemma 33 Therefe one of the fractions say h k satisfies the inequality τ h k < 5k The proof can e done similarly f the case x y < τ < x+u y+v < u v Using the process aove we can prove the theem f τ > Thus f any irrational numer τ which is etween two consecutive fractions a a where one of the fractions is the center of an Fnterval say Pm the other one is P m integer n there exists infinitely many fractions h k P such that (3) holds

9 APPROXIMATION TO IRRATIONALS VIA FAREY-PELL FRACTIONS 9 Now we will investigate that 5 is the est possile constant Let τ = + Since < 70 9 < τ < < 5 we have τ h k < 5k Here = so we otain that τ h k < 5k f infinitely many h k P Thus from Hurwitz Theem [5] we can t enlarge 5 We have seen in Theem 34 that 5 is the est possile constant over the interval (0 ) Now the question arises that Is it the est possile ( constant ) f every interval ( ) f i = 3 4? The following theems give the answer of this question Theem 35 Suppose that τ is an irrational numer etween two consecutive fractions a a in the Fnterval [P i ] where one of the fractions is the center of this Fnterval say Pm the other one is Pm Then f every integer n there exist infinitely many fractions h k P such that τ h k < 5k Further 5 is the est possile constant f the assertion Proof The existence follows y Theem 34 Let s show that the assertion fails when we replace 5 y a igger constant To prove this it is enough to show f n = Take the interval [ ] By assumption i 3 Since there is no fraction etween Pi Pi So we consider the interval in the FP [ interval ] [ ] τ can not e etween Pi Pi Pi Let [ Pi X = P ] { i Pi X = + P } i 3 Pi We fm X r+ y taking all fractions the mediants of consecutive fractions in X r Let Y = X r Now let r= X = [ 5 ] X = { } Fm X r+ y taking all fractions the mediants of consecutive fractions in X r let Y = X r Then we have a one to one crespondence etween X r X r as follows: r= a + 5 a + a + a +

10 0 ILKER AKKUS NURETTIN IRMAK AND GONCA KIZILASLAN Thus the distance etween the consecutive rationals in X r is times the distance etween consecutive rationals in X r F α 0 = + α = let + α = α0 i = α Then α is an irrational numer etween two consecutive fractions in X r Assume that there exists infinitely many pr X with τ p r < where β > 5 r = βq r then there exists infinitely many p r Y such that τ p r < where β > 5 r = βq r But this contradicts with Hurwitz s theem in [5] pp 6 Hence if we replace 5 y a larger constant the theem fails Theem 36 Suppose that τ is an[ irrational ] numer etween two consecutive fractions a a in the Fnterval where one of the fractions is the center of this Fnterval say Pm the other one is Pm Then f every integer n there exist infinitely many fractions h k P such that τ h k < 5k [ Proof Consider the interval Let τ e an irrational numer etween two ] consecutive fractions a a [ ] where one of the fractions is the center of the FP interval say Pm the other one is Pm in this interval Then we have seen that there exists infinitely many rationals h k P such that τ h k < 5k Let X = [ ] { X 3 = + We fm X r+ y taking all fractions with mediants of consecutive fractions in X r Let Y = X r Then there exists a one to one crespondence etween Y r= Y ie a a Let α = α suppose that there exists infinitely many pr α p r < If βq r α = p r + δ qr then δ < /β this implies that f infinitely many qr p r Y α p r < P ( i ) β p r + δpi (p r ) } P such that

11 APPROXIMATION TO IRRATIONALS VIA FAREY-PELL FRACTIONS F any γ > δ with 5 < γ < β we otain that f infinitely many qr p r Y α < p r γp r which contradicts with Theem 35 This proves Theem 36 Finally we give an approximation f some irrational numers τ [ ] Theem 37 Suppose that τ is an irrational numer in the interval Pi Pi [ ] Then f every integer n there exist infinitely many fractions h k P such that τ h k < 5k Proof The proof can e done similar to the proof of Theem 34 References [] D Aiylam Modified Stern-Brocot Sequences Mass Academy of Math Science January [] K Alladi Farey Sequence of Fionacci Numers The Fionacci Quarterly Vol 3 No (975) 0 [3] K Alladi Approximation of Irrationals with Farey Fionacci Irrationals The Fionacci Quarterly Vol 3 No 3 (975) [4] F Matyas A Generalization of the Farey-Fionacci Sequence Proc 7th Reg Sci Sess Math (990) [5] I Niven Diophantine Approximations Interscience Pulishers John Wiley & Sons Inc 963 Ilker Akkus Gonca Kızılaslan Department of Mathematics Faculty of Arts Science Kırıkkale University TR Kırıkkale Turkey address: iakkustr@gmailcom goncakizilaslan@gmailcom Nurettin Irmak Department of Mathematics Faculty of Science Arts Nigde University TR-545 Nigde Turkey address: nirmak@nigdeedutr irmaknurettin@gmailcom