METRIC PROPERTIES OF N-CONTINUED FRACTIONS

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1 METRIC PROPERTIES OF -COTIUED FRACTIOS DA LASCU Communicated by Marius Iosifescu A generalization of the regular continued fractions was given by Burger et al. in 2008 [3]. In this paper, we give metric properties of this expansion. For the transformation which generates this expansion, its invariant measure and Perron-Frobenius operator are investigated. AMS 200 Subject Classification: J70, K50. Key words: continued fractions, invariant measure, Perron-Frobenius operator.. ITRODUCTIO The modern history of continued fractions started with Gauss who found a natural invariant measure of the so-called regular continued fraction or Gauss transformation, i.e., T : [0, ] [0, ], T x /x /x, x 0, and T 0 0. Here denotes the floor or entire function. Let G be this measure which is called Gauss measure. The Gauss measure of an interval A B [0,] is GA / log 2 A /x + dx, where B [0,] denotes the σ- algebra of all Borel subsets of [0, ]. This measure is T -invariant in the sense that GT A GA for any A B I. By the very definition, any irrational 0 < x < can be written as the infinite regular continued fraction x a + a 2 + a : [a, a 2, a 3,...], where a n + : {, 2, 3,...} [5]. Such a n s are called incomplete quotients or continued fraction digits of x and they are given by the formulas a x /x and a n+ x a T n x, where T n denotes the nth iterate of T. Thus, the continued fraction representation conjugates the Gauss transformation and the shift on the space of infinite integer-valued sequence a n n +. Other famous probabilists like Paul Lévy and Wolfgang Doeblin also contributed to what is nowadays called the metric theory of continued fractions. MATH. REPORTS 969, 2 207, 65 8

2 66 Dan Lascu 2 The first problem in the metric theory of continued fractions was Gauss famous 82 problem [2]. In a letter dated 82, Gauss asked Laplace how fast λt n [0, x] converges to the invariant measure G[0, x], where λ denotes the Lebesgue measure on [0, ]. Gauss question was answered independently in 928 by Kuzmin [0], and in 929 by Paul Lévy [3]. Apart from regular continued fractions, there are many other continued fraction expansions: Engel continued fractions, Rosen expansions, the nearest integer continued fraction, the grotesque continued fractions, f-expansions etc. For most of these expansions the Gauss-Kuzmin-Lévy theorem has been proved [7, 9, 2, 5 9] The purpose of this paper is to show and prove some metric properties of -continued fraction expansions introduced by Burger et al. [3]. In Section 2, we present the current framework. ext we show a Legendretype result and the Brodén-Borel-Lévy formula by using the probability structure of a n n + under the Lebesgue measure. In Section 6, we find the invariant measure G of T the transformation which generate the -continued fraction expansions. In Section 7, we consider the so-called natural extension of [0,],B [0,],G,T [5]. In Section 8, we derive its Perron-Frobenius operator under different probability measures on [0, ], B [0,]. Especially, we derive the asymptotic behavior for the Perron-Frobenius operator of [0, ], B [0,], G, T. 2. -COTIUED FRACTIO EXPASIOS AS DYAMICAL SYSTEM In this paper, we consider a generalization of the Gauss transformation. Fix an integer. In [3], Burger et al. proved that any irrational 0 < x < can be written in the form 2 x a + a 2 + a : [a, a 2, a 3,...], where a n s are non-negative integers. We will call 2 the -continued fraction expansion of x. This continued fraction is treated as the following dynamical systems. Definition 2.. Fix an integer. i The measure theoretical dynamical system I, B I, T is defined as follows: I : [0, ], B I denotes the σ algebra of all Borel subsets of I, and T is the transformation

3 3 Metric properties of -continued fractions 67 x if x I, x 3 T : I I; T x : 0 if x 0. ii In addition to i, we write I, B I, G, T as I, B I, T with the following probability measure G on I, B I : dx 4 G A : log + A x +, A B I. Define the quantized index map η : I : + {0} by if x 0, x 5 ηx : if x 0. By using T and η, the sequence a n in 2 is obtained as follows: 6 a n η T n x, n with T 0 x x. Since x 0, we have that a n for any n. In this way, T gives the algorithm of -continued fraction expansion which is an obvious generalization of the regular continued fraction. Proposition 2.2. Let I, B I, G, T be as in Definition 2.ii. i I, B I, G, T is ergodic. ii The measure G is invariant under T. Proof. See [4] and Section 6. By Proposition 2.2ii, I, B I, G, T is a dynamical system in the sense of Definition 3..3 in []. 3. SOME ELEMETARY PROPERTIES OF -COTIUED FRACTIOS Roughly speaking, the metrical theory of continued fraction expansions is the asymptotic analysis of incomplete quotients a n n + and related sequences [8]. First, note that in the rational case, the continued fraction expansion 2 is finite, unlike the irrational case, when we have an infinite number of digits. In [20], Van der Wekken showed the convergence of the expansion. For x I \ Q, define the n-th order convergent [a, a 2,..., a n ] of x by truncating the expansion on the right-hand side of 2, that is, 7 [a, a 2,..., a n ] x, n.

4 68 Dan Lascu 4 To this end, for n +, define integer-valued functions p n x and q n x by 8 9 p n x : a n p n + p n 2, n 2 q n x : a n q n + q n 2, n with p 0 x : 0, q 0 x :, p x :, q x : 0, p x :, q x : a. By induction, we have 0 p n xq n x p n xq n x n, n. By using 8 and 9, we can verify that x p nx + T n xp n x q n x + T n xq n x, n. By taking T n x 0 in, we obtain [a, a 2,..., a n ] p n x/q n x. From this and by using 0 and, we obtain 2 x p nx q n x n T n x q n x T n xq, n. n x + q n x ow, since T n x < and T n xq n x + q n x, we have 3 x p nx q n x < n qnx 2, n. In order to prove 7, it is sufficient to show the following inequality: 4 x p nx q n x n, n. From 9, we have that q n x > q n x and because q 0 we have q n x > n. Finally, 4 follows from DIOPHATIE APPROXIMATIO Diophantine approximation deals with the approximation of real numbers by rational numbers [5]. We approximate x I \ Q by incomplete quotients in 8 and 9. For x I \Q, let a n be as in 6. For any n + and i n i,..., i n n, define the fundamental interval associated with i n by 5 I i n {x I \ Q : a k x i k for k,..., n} where we write I i 0 I \ Q. Remark that I i n is not connected by definition. For example, we have 6 I i {x I \ Q : a i} I \ Q i +, for any i. i

5 5 Metric properties of -continued fractions 69 Lemma 4.. Let λ denote the Lebesgue measure. Then 7 λ I i n n q n xq n x + q n x, where q n is as in 9. Proof. From the definition of T and, we have 8 I i n I \ Q ui n, vi n, where both u i n and v i n are rational numbers defined as p n x + p n x if n is odd, 9 u i n q n x + q n x : p n x if n is even, q n x and 20 v i n : By using 0, we have 7. p n x q n x p n x + p n x q n x + q n x if n is odd, if n is even. We now give a Legendre-type result for -continued fraction expansions. For x I \ Q, we define the approximation coefficient Θ x, n by 2 Θ x, n : q2 n n x p n, n where p n /q n is the nth continued fraction convergent of x in 2. Proposition 4.2. For x I \ Q and an irreducible fraction 0 < p/q <, assume that p/q is written as follows: p 22 q [i,..., i n ] where [i,..., i n ] is as in 7, and the length n + of -continued fraction expansion of p/q is chosen in such a way that it is even if p/q < x and odd otherwise. Then q p 23 Θ x, n < if and only if is the nth convergent of x q + q n q q n

6 70 Dan Lascu 6 where Θ x, n is as in 2 and the positive integer q n is defined as the denominator of the irreducible fraction representation of the rational number [i,..., i n ] with q 0 for the sequence i,..., i n. Proof. Fix x I \ Q and n. Let Θ : Θ x, n. Assume that p/q is the nth convergent of x. By 2 and the definition of Θ, we have 24 Θ q2 n x p q T n xq q + T n xq n x where we use q n q n x. Conversely, q 25 if Θ <, then q q + q n x p q < If n is even, then x > p/q and we have 26 x p q < n qq + q n. From these, 27 p q < x < p q + n qq + q n p + p n q + q n q q + q n n q + q n. where p n is defined as p n /q n [i,..., i n ]. Hence x I i n, i.e., p/q [i,..., i n ] is a convergent of x. The case when n is odd is treated similarly. 5. BRODÉ-BOREL-LÉVY FORMULA AD ITS COSEQUECES We derive the so-called Brodén-Borel-Lévy formula [6,8] for -continued fraction expansion. For x I, let a n and q n be as in 6 and 9, respectively. We define s n n 0 by 28 s 0 : 0, s n : q n q n, n. From 9, s n /a n + s n for n. Hence 29 s n a n + for n. a n a [a n, a n,..., a 2, a ],

7 7 Metric properties of -continued fractions 7 Proposition 5. Brodén-Borel-Lévy formula. Let λ denote the Lebesgue measure on I. For any n +, the conditional probability λt n < x a,..., a n is given as follows: 30 λt n < x a,..., a n s n + x s n x +, where s n is as in 28 and a,..., a n are as in 6. Proof. By definition, we have x I 3 λ T n < x a,..., a n λ T n < x I a,..., a n λ I a,..., a n for any n + and x I. From and 8 we have λ T n < x I a,..., a n p n p n + xp n 32 q n q n + xq n From this and 7, we have 33 n x q n q n + xq n. λ T n < x a,..., a n λ T n < x I a,..., a n λ I a,..., a n for any n + and x I. x q n + q n q n + xq n s n + x s n x + The Brodén-Borel-Lévy formula allows us to determine the probability structure of incomplete quotients a n n + under λ. Proposition 5.2. For any i and n +, we have 34 λa i ii +, λ a n+ i a,..., a n V,i s n where s n is as in 28, and 35 V,i x : x + x + i x + i +. Proof. From 6, the case λa i holds. For n and x I \ Q, we have T n x [a n+, a n+2,...] where a n is as in 6. By using 30, we have λ a n+ i a,..., a n λ T n i +, ] a,..., a n. i

8 72 Dan Lascu 8 36 s n + i s n i + s n + i+ s n i+ + V,is n. In 34, i λa n+ i a,..., a n must be because λ is a probability measure on I, B I. This can be verified from 34 and 36 by using the partial fraction decomposition. By the same token, we see that 37 V,i x for any x I. i Remark 5.3. Proposition 5.2 is the starting point of an approach to the metrical theory of -continued fraction expansions via dependence with complete connections see [6], Section 5.2. We apply this method in [] to obtain a solution of Gauss-Kuzmin-type problem for -continued fraction expansions. Corollary 5.4. The sequence s n n + with s 0 0 is a homogeneous I-valuated Markov chain on I, B I, λ with the following transition mechanism: from state s I the only possible one-step transitions are those to states /s+ i, i, with corresponding probabilities V,i s, i. 6. THE IVARIAT MEASURE OF T Let I, B I be as in Definition 2.i. In this section, we will give the explicit form of the invariant probability measure G of the transformation T in 3, i.e., G T A G A for any A B I. From the aspect of metric theory, the digits a n in 6 can be viewed as random variables on I, B I that are defined almost surely with respect to any probability measure on B I assigning probability 0 to the set of rationals in I. Such a probability measure is Lebesgue measure λ, but a more important one in the present context is the invariant probability measure G of the transformation T. Proposition 6.. The invariant probability density ρ of the transformation T is given by 38 ρ x k x +, x I where k is the normalized constant such that the invariant measure G is a probability measure. Furthermore, the constant k is given in 4, i.e., k log +. Proof. We will give a proof which involves properties of the Perron- Frobenius operator of T under G. Therefore, the proof will be given in Section 8.

9 9 Metric properties of -continued fractions ATURAL EXTESIO AD EXTEDED RADOM VARIABLES Fix an integer. In this section, we introduce the natural extension T of T in 3 and its extended random variables [5]. Let I, B I, T be as in Definition 2.i. Define u,i i by 39 u,i : I I; u,i x : x + i, x I. For each i, u,i is a right inverse of T, that is, 40 T u,i x x, for any x I. Furthermore, if ηx i, then u,i T x x where η is as in 5. Definition 7.. The natural extension I 2, B I 2, T of I, B I, T is the transformation T of the square space I 2, B 2 I : I, B I I, B I defined as follows [4]: 4 T : I 2 I 2 ; T x, y : T x, u,ηx y, x, y I 2. From 40, we see that T is bijective on I 2 with the inverse 42 T x, y u,ηy x, T y, x, y I 2. Iterations of 4 and 42 are given as follows for each n 2: T n x, y T n x, [x n, x n,..., x 2 x, x + y], 43 T n x, y [yn, y n,..., y 2, y + x], T n y 44 where x i : η T i x and y i : η T i y for i,..., n. For G in 4, Dajani et al. [4] define its extended measure G on I 2, BI 2 as 45 G B : log dxdy + B xy + 2, B B2 I. Then G A I G I A G A for any A B I. The measure G is preserved by T [4]. Define the projection E : I 2 I by Ex, y : x. With respect to T in 4, define extended incomplete quotients a l x, y, l Z at x, y I 2 by 46 a l x, y : η E T l x, y, l Z. Remark 7.2. i Remark that a l x, y in 46 is also well-defined for l 0 because T is invertible. By 43 and 44, we have 47 a n x, yx n, a 0 x, yy, a n x, yy n+, n +, x, y I 2 where we use notations in 43 and 44.

10 74 Dan Lascu 0 ii Since the measure G is preserved by T, the doubly infinite sequence a l x, y l Z is strictly stationary i.e., its distribution is invariant under a shift of the indices under G. Theorem 7.3. Fix x, y I 2 and let a l : a l x, y for l Z. Define a : [a 0, a,...]. Then the following holds for any x I: + ax 48 G [0, x] I a 0, a,... G -a.s. ax + Proof. Recall fundamental interval in 5. Let I,n denote the fundamental interval I a 0, a,..., a n for n. We have 49 G [0, x] I a 0, a,... lim n G [0, x] I a 0,..., a n and 50 G [0, x] I a 0,..., a n G [0, x] I,n G I I,n k m dy G I,n I,n G I,n xy n + xy n + x 0 xy + I,n xy + for some y n I,n where k m is as in Proposition 6.. Since 5 lim n y n [a 0, a,...] a, the proof is completed. du yu + 2 G dy The stochastic property of a l l Z under G is given as follows. Corollary 7.4. For any i, we have 52 G a i a 0, a,... V,i a G -a.s. where a [a 0, a,...] and V,i is as in 35. Proof. Let I,n be as in the proof of Theorem 7.3. We have 53 G a i a 0, a,... lim n G a i I,n. ow G i +, i ] [0, I,n ] G i+, i I,n G I I,n G -a.s.

11 Metric properties of -continued fractions 75 V,i y G dy G I,n I,n 54 V,i y n for some y n I,n. From 5, the proof is completed. Remark 7.5. The strict stationarity of a l l Z, under G implies that 55 G a l+ i a l, a l,... V,i a G -a.s. for any i and l Z, where a [a l, a l,...]. The last equation emphasizes that a l l Z is a chain of infinite order in the theory of dependence with complete connections [6]. 8. PERRO-FROBEIUS OPERATORS Let I, B I, G, T be as in Definition 2.ii. In this section, we derive its Perron-Frobenius operator. Let µ be a probability measure on I, B I such that µ T A 0 whenever µa 0 for A B I. For example, this condition is satisfied if T is µ-preserving, that is, µt µ. Let L I, µ : {f : I C : I f dµ < }. The Perron-Frobenius operator of I, B I, µ, G is defined as the bounded linear operator U on the Banach space L I, µ such that the following holds: 56 Uf dµ f dµ for all A B I, f L I, µ. A T A For more details, see [, 8] or Appendix A in [2]. Proposition 8.. Let I, B I, G, T be as in Definition 2.ii, and let U denote its Perron-Frobenius operator. Then: i The following equation holds: 57 {Uf}x V,i x f, f L I, G, x + i i where V,i is as in 35. ii Let µ be a probability measure on I, B I such that µ is absolutely continuous with respect to the Lebesgue measure λ and let h : dµ/dλ a.e. in I. Then the following holds: a Let S denote the Perron-Frobenius operator of T under µ. Then the following holds a.e. in I:

12 76 Dan Lascu {Sf}x h x+i hx x + i 2 f x + i i {U ˆf}x x + hx for f L I, µ, where ˆfx : x+fxhx, x I. In addition, the nth power S n of S is written as follows: 60 {S n f}x {U n ˆf}x x + hx for any f L I, µ and any n. b Let K denote the Perron-Frobenius operator of T under λ. Then the following holds a.e. in I: 6 {Kf}x x + i 2 f, f L I, λ. x + i i In addition, the nth power K n of K is written as follows: 62 {K n f}x {U n ˆf}x x +, f L I, λ, for any f L I, λ and any n, where ˆfx : x + fx, x I. c For any n + and A B I, we have 63 µ T n A {U n f}xdg x where fx : log + A x + hx for x I. Proof. ] i Let T,i denote the restriction of T to the subinterval I i : i+, i, i, that is, 64 T,i x x, x I i. Let CA : T A and C i A : T,i A for A B I. Since CA i C ia and C i C j is a null set when i j, we have 65 f dg f dg, f L I, G, A B I. CA i C i A

13 3 Metric properties of -continued fractions 77 For any i, by the change of variables x T,i y successively obtain fx G dx 66 C i A + log + log V,i y f y + i A C i A A fx + x dx f y+i + y+i G dy. ow, 57 follows from 65 and 66. iia From 64, for any f L I, G and A B I, we have 67 fx µdx fx µdx. CA i C i A Then 68 C i A fxµdx C i A From 67 and 68, 69 fx µdx CA fxhx dx f A i A f y + i h x + i x + i y + i 2 dy y + i, we h y + i y + i 2 dy. x + i 2 dx. Since dµ hdx, 58 follows from 69. ow, since ˆfx x + fxhx, from 58, we have 70 {U ˆf}x x + h x+i x + i 2 f. x + i i From 58 and 70, 59 follows immediately. iib The formula 6 is a consequence of 59 and follows immediately. iic We will use mathematical induction. For n 0, the equation 63 holds by definitions of f and h. Assume that 63 holds for some n. Then 7 µ T n+ CA A µ T n T A A CA {U n f}x G dx. Since U U T and 56, we have 72 {U n f}x G dx {U n+ f}x G dx.

14 78 Dan Lascu 4 Therefore, 73 µ which ends the proof. T n+ A {U n+ f}xg dx A For a function f : I C, define the variation var A f of f on a subset A of I by k 74 var A f : sup ft i+ ft i i where the supremum being taken over t < < t k, t i A, i k, and k 2 [8], p. 75. We write simply varf for var I f. Let BV I : {f : I C : var f < } and let L I denote the collection of all bounded measurable functions f : I C. It is known that BV I L I L I, µ. Let LI denote the Banach space of all complex-valued Lipschitz continuous functions on I with the following norm L : 75 f L : sup fx + sf, x I with 76 sf : sup x y fx fy, f LI. x y In the following proposition we show that the operator U in 57 preserves monotonicity and enjoys a contraction property for Lipschitz continuous functions. Proposition 8.2. Let U be as in 57. i Let f L I. Then the following holds: a If f is non-decreasing non-increasing, then U f is non-increasing non-decreasing. b If f is monotone, then 77 var Uf + varf. ii For any f LI, we have 78 suf q sf, where 79 q : i i 3 i + + i + ii + 3. Proof. ia To make a choice assume that f is non-decreasing. Let x < y, x, y I. We have {Uf}y {Uf}x S + S 2, where

15 5 Metric properties of -continued fractions S V,i y f f y + i i S 2 i m V,i y V,i x f x + i x + i., Clearly, S 0. ow, since i V,ix for any x I, we can write 82 S 2 f f V,i y V,i x. x + x + i i As is easy to see, the functions V,i are increasing for all i. Also, using that f x+ f x+i and the proof is complete. 83 var Uf {Uf}0 {Uf} i, we have that S 2 0. Thus {Uf}y {Uf}x 0 ib Assume that f is non-decreasing. Then by a we have V,i 0f i By calculus, we have V,i f var Uf ii + f + i i + i + 2 f i + i + f i + i + 2 f i + i i V,i y i + i + 2 f0 + f i f f0 + + varf. ii For x y, x, y I, we have {Uf}y {Uf}x V,i y V,i x f y x y x x + i i f y+i f x+i 84 Remark that x+i x+i 85 V,i u i + u + i + + i u + i, i, and then V,i y V,i x f y x x + i i. + i. x + i y + i

16 80 Dan Lascu i i + f y + i + x + i + x + i + f. x + i Assume that x > y. It then follows from 86 and 84 that {Uf}y {Uf}x y x sf i + y + iy + i V,iy y + i 2 i where q is as in 79. Since q sf 88 suf sup then the proof is complete. x,y I,x y {Uf}y {Uf}x y x Proof of Proposition 6.. For I, B I, G, T in Definition 2.ii, let U denote its Perron-Frobenius operator. Let 89 ρ x : k x +, x I, where k log +. From properties of the Perron-Frobenius operator, it is sufficient to show { that the function ρ } defined in 89 satisfies Uρ ρ. Since T x x + i, i, x I, we have 90 {Uρ }x d dx i T [0,x] ρ t dt x + i 2 ρ By definition of ρ, we see that 9 {Uρ }x i x + i t T x. x + ix + i + ρ x. ρ t T t Hence the statement is proved. REFERECES [] A. Boyarsky and P. Góra, Laws of Chaos: Invariant Measures and Dynamical Systems in One Dimension. Birkhäuser, Boston, 997. [2] C. Brezinski, History of Continued Fractions and Padé Approximants. Springer Series in Computational Mathematics 2, Springer-Verlag, Berlin, 99.

17 7 Metric properties of -continued fractions 8 [3] E.B. Burger, J. Gell-Redman, R. Kravitz, D. Walton and. Yates, Shrinking the period lengths of continued fractions while still capturing convergents. J. umber Theory ,, [4] K. Dajani, C. Kraaikamp and. Van der Wekken, Ergodicity of -continued fraction expansions. J. umber Theory , 9, [5] A.Ya. Khinchin, Continued Fractions. Univ. Chicago Press, Chicago, 964 [Translation of the 3rd Russian Edition, 96]. [6] M. Iosifescu and S. Grigorescu, Dependence With Complete Connections and its Applications. Cambridge Tracts in Mathematics 96, Cambridge Univ. Press, Cambridge, 2nd edition, [7] M. Iosifescu and S. Kalpazidou, The nearest integer continued fraction expansion: an approach in the spirit of Doeblin. Contemp. Math , [8] M. Iosifescu and C. Kraaikamp, Metrical Theory of Continued Fractions. Kluwer Academic Publishers, Dordrecht, [9] M. Iosifescu and G.I. Sebe, An exact convergence rate in a Gauss-Kuzmin-Lévy problem for some continued fraction expansion. In: Mathematical Analysis and Applications, pp AIP Conf. Proc. 835, [0] R.O. Kuzmin, On a problem of Gauss. Dokl. Akad. auk SSSR Ser. A [Russian; French version in Atti Congr. Internaz.Mat. Bologna, 928, Tomo VI 932, Zanichelli, Bologna]. [] D. Lascu, Dependence with complete connections and the Gauss-Kuzmin theorem for -continued fractions. J. Math. Anal. Appl , [2] D. Lascu, On a Gauss-Kuzmin-type problem for a family of continued fraction expansions. J. umber Theory , 7, [3] P. Lévy, Sur les lois de probabilité dont dépendent les quotients complets et incomplets d une fraction continue. Bull. Soc. Math. France , [4] H. akada, Metrical theory for a class of continued fraction transformations and their natural extensions. Tokyo J. Math. 4 98, 2, [5] F. Schweiger, Ergodic theory of fibred systems and metric number theory. Clarendon Press, Oxford, 995. [6] G.I. Sebe, A two-dimensional Gauss-Kuzmin theorem for singular continued fractions. Indag. Mathem..S. 2000, 4, [7] G.I. Sebe, On convergence rate in the Gauss-Kuzmin problem for grotesque continued fractions. Monatsh. Math , [8] G.I. Sebe, A Gauss-Kuzmin theorem for the Rosen fractions. J. Théor. ombres Bordeaux , [9] G.I. Sebe and D. Lascu, A Gauss-Kuzmin theorem and related questions for θ-expansions. Journal of Function Spaces , 2 pages. [20] C.D. Van der Wekken, Lost periodicity in -continued fraction expansions. Bachelor thesis, Delft University of Technology, Delft, 20, uuid:6737fff-f3e3-44e4-8e59-5e e/. Received 7 October 205 Mircea cel Batran aval Academy, Fulgerului, Constanta, Romania lascudan@gmail.com