Arithmetic and metric properties of Oppenheim continued fraction expansions

Transcription

1 Journal of Number Theory 27 ( Arithmetic and metric properties of Oppenheim continued fraction expansions Ai-Hua Fan a,b,, Bao-Wei Wang b,c,junwu c a Department of Mathematics, Wuhan University, Wuhan , PR China b CNRS UMR 640-LAMFA, Université de Picardie, Amiens, France c Department of Mathematics, Huazhong University of Science and Technology, Wuhan , PR China Received March 2006; revised 2 November 2006 Available online 2 March 2007 Communicated by David Goss Abstract We introduce a class of continued fraction expansions called Oppenheim continued fraction (OCF expansions. Basic properties of these expansions are discussed and metric properties of the digits occurring in the OCF expansions are studied Elsevier Inc. All rights reserved. MSC: primary K55; secondary J70 Keywords: Oppenheim series expansion; Oppenheim continued fraction expansion; Metric property. Introduction Over the last twenty years, considerable interests are shown in various continued fraction expansions. Examples of such continued fraction expansions include, for example, the backward continued fraction [], the fraction with even partial quotients [8] and the Farey-shift [9]. For more details, see [8] and the references therein. Metric properties of these expansions have been well studied. Each of these continued fraction expansions is ergodic and has an infinite and σ -finite, invariant measure which is absolutely continuous with respect to Lebesgue measure. * Corresponding author. addresses: ai-hua.fan@u-picaride.fr (A.-H. Fan, bwei_wang@yahoo.com.cn (B.-W. Wang, wujunyu@public.wh.hb.cn (J. Wu X/$ see front matter 2007 Elsevier Inc. All rights reserved. doi:0.06/j.jnt

2 A.-H. Fan et al. / Journal of Number Theory 27 ( In 2002, Y. Hartono, C. Kraaikamp and F. Schweiger [7] introduced a new continued fraction algorithm with nondecreasing partial quotients, named Engel continued fraction (ECF expansion. Just as the name suggests, the ECF expansion is originated from the classical Engel series expansion. Recall that the Engel series expansion is generated by the transformation S E :[0, [0, given by S E (x := ([ ] ( + x x [ x ]+, x 0; S(0 := 0, (. where [ξ] denotes the largest integer not exceeding ξ. While the Engel continued fraction expansion is generated by the transformation T E : [0, [0, given by ( ( [ ] T E (x := [ x ] x, x 0; T E (0 := 0. (.2 x For each x (0,, the transformation S E generates a (unique series expansion of the form x = d (x + d (xd 2 (x + + d (xd 2 (x d n (x +, where d n (x =[/S n (x]+, n, and the digits {d n (x: n } satisfy the condition 2 d n (x d n+ (x for all n. In fact, it was W. Sierpiński [6] who first studied these series expansions in 9. Metric properties of Engel series expansion were established by P. Erdös, A. Rényi and P. Szüsz [2] and A. Rényi [5]. F. Schweiger [7] showed that S E is ergodic and Thaler [20] found a whole family of infinite and σ -finite measure for S E. Fractal properties of exceptional sets related to the Engel series expansion have been discussed by Y.Y. Liu and J. Wu [2]. For each x (0,, the transformation T E generates a new type of continued fraction expansion of the form x = d (x + d (x d 2 (x+...+ d n (x dn(x+... where d n (x =[/TE n (x]+, n, and the digits {d n(x: n } satisfy the condition d n (x d n+ (x for all n. Arithmetic and ergodic properties of T E associated to this new continued fraction expansion were studied by Y. Hartono, C. Kraaikamp and F. Schweiger in [7]. They showed that T E has no finite invariant measure equivalent to the Lebesgue measure, but has infinitely many σ -finite, infinite invariant measures. Also they showed that T E is ergodic with respect to Lebesgue measure. In [], C. Kraaikamp and J. Wu derived some metric properties of the digits {d n (x: n } occurring in such an expansion. They also investigated different kinds of exceptional sets on which the metric properties fail to hold. Even if the ergodic and metric properties of both Engel series and Engel continued fraction are very similar, F. Schweiger [9] constructed a class of algorithms, with proper choice of a parameter, with increasing digits which have quite different properties from those of Engel case.,

3 66 A.-H. Fan et al. / Journal of Number Theory 27 ( In the present paper, we introduce a class of continued fraction expansions, which we call Oppenheim continued fraction (OCF expansions. The Engel continued fraction (ECF expansion stands as a special case. In general, there is no longer an associate dynamical system. We will use probability method to study these OCFs. Before introducing the Oppenheim continued fraction expansions, we would like to present Oppenheim series expansions [6]. Let {γ j } j be a sequence of positive rational-valued functions defined on N \{} satisfying γ j (n, for all j. n(n For each real number 0 <x<, define the integers d j = d j (x and the real numbers x j (j =, 2,...by the algorithm: [ ] x = x, d j = +, x j+ = (x j dj. (.3 x j γ j (d j This leads to the Oppenheim series expansion of x: Write /d + γ (d /d 2 + γ (d γ 2 (d 2 /d 3 +. (.4 h j (n = γ j (nn(n. If h j is integer-valued for all j, the series (.4 is said to be a restricted Oppenheim series expansion of x. Here are some special cases which were extensively studied: Lüroth series expansion: h j (n = ; Engel series expansion: h j (n = n ; Sylvester series expansion: h j (n = n(n. The expansion (.4 with d j defined by (.3 was first studied by A. Oppenheim [4] who established basic arithmetical properties, including the criteria of rationality of the expansion. The metric theory for Oppenheim series expansion was established by J. Galambos [3 5]. Exceptional sets associated with this expansion were discussed by J. Wu [22,23]. Further information on the Oppenheim series expansion can be found in J. Galambos [3,6], F. Schweiger [8] and W. Vervaat [2]. Now, let us introduce Oppenheim continued fraction expansion. Let {h j } j be a sequence of nonnegative integer valued functions defined on N. For any x (0,, define the integers d j = d j (x and the real numbers x j, j =, 2,..., by the algorithm: [ ] ( x = x, d j =, x j+ = d j. (.5 x j h j (d j + x j It can be proved that this algorithm leads to a kind of continued fraction expansion of x (0, with the following form

4 A.-H. Fan et al. / Journal of Number Theory 27 ( x = d + h (d + d h n (d n + dn+(hn(dn+x n+ (.6 where d n N and d n+ h n (d n + for all n (see Proposition 2.5. Here we list some special cases of Oppenheim continued fraction expansion: Regular continued fraction expansion: h j (n = 0; Engel continued expansion: h j (n = n ; Sylvester continued expansion: h j (n = n(n. In this paper, we discuss basic arithmetic properties (Section 2 and study metric properties of the digits {d n (x: n } occurring in these expansions, including weak and strong large number law and the central limit theorem (Section 3. We also investigate approximation speed (Section OCF expansions 2.. Some arithmetic properties In this subsection, we study the arithmetic properties, including the convergence theorem of Oppenheim continued fraction expansions, uniqueness of the expansion, and a general property on the digits. Definition 2.. A vector of positive integers (d,d 2,...,d n is called an admissible vector for the Oppenheim continued fraction expansion if there exists x (0, such that d j (x = d j for all j n. A sequence (d,d 2,...,d n,...is called an admissible sequence if (d,d 2,...,d n is an admissible vector for each n. Proposition 2.2. A sequence of positive integers (d,d 2,...,d n,... is admissible if and only if for each j, d j+ h j (d j +. (2. Proof. The necessity of (2. is obvious by the definition of d j = d j (x. To prove the sufficiency, for each n, we take Since x = x = d + h (d + d h n (d n + dn d + <x < d, then d (x = d. Hence, by the algorithm (.5, we know x 2 = d 2 + h 2 (d 2 + d h n (d n + dn.. By induction, we get d j (x = d j for all j n.

5 68 A.-H. Fan et al. / Journal of Number Theory 27 ( The following proposition is a characterization of rational numbers. Proposition 2.3. A number x (0, has a finite Oppenheim continued fraction expansion (i.e., x j = 0 for some j if and only if x Q. Proof. By the expansion of x in (.6, it is necessary that x Q if x has a finite expansion. Suppose now x is rational. By the algorithm, we know if x j 0, then x j is rational and 0 <x j <, hence x j := a j a b j = j d j a j +a j+ where 0 a j+ <a j and d j =[b j /a j ]. Thus by the algorithm, we have ( a j+ x j+ = d j = := a j+. h j (d j + x j h j (d j + a j b j+ Because a j+ <a j, then this procedure will stop at finite steps, that is to say, x j = 0for some j. The proof of the following proposition is omitted, since it is quite straightforward. For a proof the interested reader is referred to Section in [0] where a similar result has been obtained for a class of continued fractions. Proposition 2.4. Let {a n }, {b n } be two sequences of positive numbers. Let {p n } n and {q n } n be the sequences recursively defined by Then one has p 0 = 0, p =, p n = b n p n + a n p n 2, for n 2, (2.2 q 0 =, q = b, q n = b n q n + a n q n 2, for n 2. (2.3 (i p n q n p n q n = ( n n j= a j. (ii b + a b a n bn = p n q n. (iii For {p n /q n } n, even-numbered terms are strictly increasing and odd-numbered strictly decreasing, moreover, every even-numbered term is less than every odd-numbered one. As for the usual continued fractions, we get the convergents obtained through finite truncation: for each n, let Q n (d,...,d n = d + h (d + d h n (d n + dn For simplicity and convenience, we denote Q n (d,...,d n ; 0 = Q n (d,...,d n and for any 0 <x, define Q n (d,...,d n ; x by replacing d n in Q n (d,...,d n by d n + x. From Proposition 2.4, applied to b n = d n and a n = h n (d n +, we get Q n (d,...,d n = p n q n..

6 A.-H. Fan et al. / Journal of Number Theory 27 ( If d j = d j (x, we call p n(x q n (x the nth convergent of x in its Oppenheim continued fraction expansion. The sequence of convergent converges to the number from which it is generated, as the following proposition shows. Proposition 2.5. For every x (0,, we have p n (x lim = x. (2.4 n + q n (x Proof. If x is rational, we conclude (2.4 by Proposition 2.2. Now let x be irrational. By Proposition 2.4(ii, one has Hence following from Proposition 2.4(i, we get x = p n(x + (h n (d n (x + x n+ p n (x q n (x + (h n (d n (x + x n+ q n (x. (2.5 x p n nj= q = (h j (d j (x + x n+ n q n (x(q n (x + (h n (d n (x + x n+ q n (x nj= ( (h j (d j (x + q n (x 2 for x n+. (2.6 d n+ (x d n+ (x On the other hand, by (2.3, q n d n q n n k= d k. Moreover, (d,...,d n is admissible, then x p n(x q n (x q n (x n j= h j (d j (x + d j+ (x q n (x 0. The last assertion in the above formula is followed by the fact that q n q n + q n 2. The next proposition shows that the Oppenheim continued fraction expansion is unique. Proposition 2.6. Let (d,...,d n,...be admissible, and {p n } n and {q n } n be given by (2.2 and (2.3 with b n = d n and a n = h n (d n +. Then p n q n converges to some x (0, and d n (x = d n for all n. Proof. The existence of the limit follows from Proposition 2.2(iii and p n q n p n+ q n+ 0. Since 0 <p 2 /q 2 <x<p /q, we have x (0,. Notice that for any n, p 2n q 2n <x< p 2n+ q 2n+, thus x can be expanded with the form x = d + h (d + d h 2n (d 2n + d 2n +x

7 70 A.-H. Fan et al. / Journal of Number Theory 27 ( for some 0 <x <. Using similar arguments in the proof of Proposition 2.2, we get d j (x = d j for all j 2n Some preliminary results Here and in what follows, we use λ to denote the Lebesgue measure. Definition 2.7. Let (d,d 2,...,d n be an admissible sequence. Define B(d,d 2,...,d n := { } x (0, ]: d (x = d,d 2 (x = d 2,..., d n (x = d n which is called an nth order cylinder. For integers a b and real number 0 y, define Υ(a,b,y= Then about the measure of a cylinder, we have Proposition 2.8. Let (d,d 2,...,d n be an admissible vector. Then λ(b(d,...,d n,d n+ λ(b(d,...,d n λ ( B(d,...,d n = a( + y (b + ay(b + + ay. (2.7 n j= (h j (d j +, q n (q n + q n = Υ ( h n (d n +,d n+,y n with y n = q n q n. (2.8 Proof. Look at (.6. Since 0 (h n (d n + x n+ <, B(d,d 2,...,d n is the interval with two endpoints Q n (d,...,d n ; 0 and Q n (d,...,d n +. By Proposition 2.4, these two endpoints are p n q n and p n+p n q n +q n. As a result, λ ( B(d,d 2,...,d n = p n p n + p n n q n q n + q = j= (h j (d j +. n q n (q n + q n Our further investigations are partially based on (2.8. We are led to study the function Υ(a,b,y. Lemma 2.9. For any integers a b and any real number 0 y<, we have (i Υ(a,b,y= (ii (iii a( + y b + ay a( + y b + + ay = b++ay a(+y b+ay a(+y Υ(a,b,y Υ(a,b,0 4ay b(b + ; Υ(a,b,y 2a b 2. x 2 dx;

8 A.-H. Fan et al. / Journal of Number Theory 27 ( Proof. The proof is elementary. Lemma 2.0. For any integer a,τ N and any real number 0 y<, we have τ + b τa Proof. It suffices to remark that, by (i in Lemma 2.9, we have Υ(a,b,y 2 τ +. (2.9 Υ(a,b,y= ( a( + y b + ay b τa b τa a( + y = + y b + + ay τ + y. Lemma 2.. For any integer a N and any real number 0 y< a, we have Proof. For the first term, we have b a a Υ(a,b,y b 2. a a Υ (a,a,y = a a( + y a (a + ay(a + + ay = a + + ay a +. For other terms, notice that a( + y < a +, we have However, a b Υ(a,b,y a b a + a(a + (b + ay(b + + ay b 2 (b +. b a+2 + b 2 (b + z 2 (z + dz 2(a + 2. a+ Therefore, the sum to be estimated is bounded by h(a := a + + a (a + (a a 2(a +. Since a+ + a a (a+(a+2 is decreasing and 2(a+ is bounded by 2,wehave { h(a max h(, h(2, h(3, (4 + ( } = h( = 2 2.

9 72 A.-H. Fan et al. / Journal of Number Theory 27 ( Lemma 2.2. For any integer a N and any real number 0 y<, we have log b + a Υ(a,b,y 2 b a k= log(k + k 2 Proof. It follows just by Υ(a,b,y 2a b 2 (see Lemma 2.9(iii. < +. Lemma 2.3. For any fixed integer a N and real number 0 y<, we have, for τ sufficient large, a b τa b a Υ(a,b,y= ( + o( + O(y log τ. Proof. By Lemma 2.9(ii, we know b a Υ(a,b,y b + 4y b +. Lemma 2.4. For any integer a N and any real number 0 y< a, we have Υ(a,b,y e it log a b it b a t a. Proof. We denote by α(t,a,y the quantity defined by the sum. Lemma 2.9 gives α(t,a,y it = b++ay a(+y b a b+ay a(+y x 2 eit log a b e it log x dx. We can get the desired result since by the mean value theorem and when b+ay we have e it log a b e it log x t log b a log x = t a. 3. Metric theory of OCF expansions a(+y x b++ay a(+y, In this section, we will investigate the metric properties of the digits d j. In most cases, we follow the ideas due to J. Galambos [3 5] who studied the metric properties of Oppenheim series expansions. In the sequel, we always assume that the following hypothesis holds: h j (n n for all j and n. (H

10 A.-H. Fan et al. / Journal of Number Theory 27 ( About d j (x and x j d j Proposition 3.. Assume (H. We have λ(b(d,d 2,...,d n d d,d 2,...,d n n where the summation takes over all admissible vectors (d,d 2,...,d n. Proof. Let S n be the sum in (3.. We are going to show that ( n, (3. 2 By (2.8, we can write S n+ 2 S n. S n+ = λ(b(d,...,d n Θ(d n,y n d d,...,d n n where Θ(d n,y n := d n+ h n (d n + d n d n+ Υ ( h n (d n +,d n+,y n, and the summation in S n+ takes over all admissible vectors (d,...,d n,i.e.d j+ h j (d j +. According to Lemma 2., Θ(d n,y n is bounded by 2. So, we get S n+ 2 S n. Theorem 3.2. Assume (H. Then for almost all x (0,, d j (x is strictly increasing when j is sufficiently large. Proof. We can show: lim inf j d j+ (x d j (x for almost all x. By Proposition 2.2 and (H, we have d j+ h j (d j + d j for all x. Consider A n ={d n+ = h n (d n + }. Then λ(a n = d,...,d n λ ( B(d,...,d n,h n (d n +. Notice that Υ (a,a,y = a++ay a+ and h n(d n + d n. We get λ(b(d,...,d n,h n (d n + λ(b(d,...,d n = Υ ( h n (d n +,h n (d n +,y n d n +. Hence, by Proposition 3., we get λ(a n ( 2 n. We conclude by Borel Cantelli lemma. For every x (0, and j, define z j (x = ( ( h j dj (x + x j+ = { } d j =. (3.2 x j x j Next, we show that z j converges in distribution to the uniform distribution on [0, ].

11 74 A.-H. Fan et al. / Journal of Number Theory 27 ( Theorem 3.3. Assume (H. Then for any 0 <c, λ { x (0, : z n (x < c } c ( n c. (3.3 2 Proof. Write λ ( z n+ (x < c = λ { ( ( x (0, : x n+ hn dn (x + <c } = { ( ( x: d (x = d,...,d n (x = d n,x n+ hn dn (x + <c } d,...,d n λ := d,...,d n λ ( B(d,...,d n ; c. From (.6, it is evident that B(d,...,d n ; c is the interval with the two endpoints Q n (d,..., d n ; 0 and Q n (d,...,d n ; c. Then, by Proposition 2.4, we have λ ( B(d,...,d n ; c = p n p n + cp n q n q n + cq = c n j= (h j (d j +. n q n (q n + cq n Taking summation over d,...,d n and using Proposition 3., we get the desired result Growth rate of digits d n (x The condition d j+ h j (d j + can be viewed as a growth rate of the sequence {d n (x} n. While, for a rational number x, the digits d n (x terminate at finite steps. So, a general result for all numbers with faster growth rate is hardly to seek. For each x (0,, define R n := R n (x = d n+(x h n (d n (x +. We can consider the behavior of R n at the infinity as a growth rate of d n. Theorem 3.4. Let ϕ : N N. Then (i If n= ϕ(n < +, then λ{x (0, : R n(x > ϕ(n i.o.}=0. (ii If n= ϕ(n =+, then λ{x (0, : R n(x > ϕ(n i.o.}=. Before giving the proof, we establish a general estimation of conditional probability that will be used several times. Lemma 3.5. Let (A,...,A n,...be a sequence of nonempty subsets of N, and ϕ : N N. Then we have ϕ(n + λ{x: R (x A,...,R n (x A n,r n >ϕ(n} 2 λ{x: R (x A,...,R n (x A n } ϕ(n +.

12 A.-H. Fan et al. / Journal of Number Theory 27 ( Proof. The numerator is equal to λ { x: R (x A,...,R n (x A n,r n >ϕ(n } = λ ( B(d,...,d n,d n+ d,...,d n d n+ >ϕ(n(h n (d n + = d,...,d n λ ( B(d,...,d n d n+ >ϕ(n(h n (d n + Υ ( h n (d n +,d n+,y n where the first sum takes over all (d,...,d n satisfying R j A j. An application of Lemma 2.0 yields that the inner sum is bounded by. This finishes the proof. ϕ(n+ and 2 ϕ(n+ Proof of Theorem 3.4. Notice that Lemma 3.5 gives ϕ(n + λ{ x (0, : R n (x > ϕ(n } 2 ϕ(n +. (3.4 Therefore, Borel Cantelli lemma guarantees the first part of the theorem. For the second part, consider the sets A N (M = { x (0, : R n (x ϕ(n, N n N + M } (N, M and A = + N= M= A N (M. Then A is the set of the points x for which R n (x ϕ(n holds for all but a finite number of n. Therefore we only need to show that λ(a = 0, or equivalently, lim M λ(a N (M = 0 for all N. Since λ(a N (M = λ(a N (M A N (M + c + λ(a N (M +, by Lemma 3.5, we get ( 2 λ(a ( N(M + ϕ(n + M + + λ(a N (M. ϕ(n + M + + Therefore the divergence of n= ϕ(n implies lim M λ(a N (M = 0 for each N. Thus λ(a = 0. As a corollary, we have Theorem 3.6. For almost all x (0,, lim sup n log R n (x log n log log n =, lim inf n log R n (x log n log log n =. (3.5 Proof. Applying the preceding theorem to the two cases ϕ(n = n log n and n(log n +ɛ allows us to get immediately the first equality. To show the second one, we define A n (α = {R n (x n(log n α }, for any α>0 and n. Then by Fatou Lemma and Lemma 3.5, we get λ(lim sup n + A n (α lim sup n + λ(a n (α. This implies that for almost all x (0,

13 76 A.-H. Fan et al. / Journal of Number Theory 27 ( lim inf n log R n (x log n log log n α. This proves the second equality, for α>0 is arbitrary. For a further investigation on the ratio R n, we consider Theorem 3.7. For almost all x (0,, lim sup n log L n (x log n log log n L n = max j n R n. =, lim inf n log L n (x log n log log n = 0. (3.6 log L Proof. We only establish that lim inf n (x log n n log log n 0, since the other assertions can be done similar to Theorem 3.6 by replacing A n (α by Ã(β ={L n (x n(log n β },forβ>0. To get the desired inequality, it suffices to show that for any 0 <α<, for almost all x (0,, L n (x n(log n α := ϕ(n holds for only finite times. But by Theorem 3.4, for almost all x (0,, L n (x > ϕ(n for infinitely many n. Thus it follows that L n (x ϕ(n holds for only finite times is equivalent to L n (x ϕ(n, but L n+ (x > ϕ(n + holds for only finite times. Let A n ={x (0, : L n (x ϕ(n, L n+ (x > ϕ(n + }. Then Lemma 3.5 is enough to make sure that n= λ(a n <. Following the Borel Cantelli lemma, we complete the proof Laws of large numbers on log R n and R n In this subsection, we will study the law of large numbers (L-N of the digits occurring in the Oppenheim continued fraction expansions. We shall make use of the following more general law of large numbers (see Galambos [3]. Lemma 3.8. Let X,X 2,...,X n,...be a sequence of random variables defined on a given probability space and assume that (i lim n EX n = m, for some constant m R. (ii Var( n i= X i = O(n t for some 0 <t<2. (iii X i M(i=, 2,...,forsomeM R. Then n ni= X i converges almost surely to m. Using this lemma, we will prove the following law of large numbers for log R n.

14 A.-H. Fan et al. / Journal of Number Theory 27 ( Theorem 3.9. Assume (H. Then for almost all x (0,, lim n + ( log R (x + log R 2 (x + +log R n (x =. (3.7 n Proof. We apply Lemma 3.8 to X j = log R j. It is easy to see that X j 0 because R j. Comparing z j = (h j (d j + x j+ with the definition of R j, we get ( X j = log z j + O d j+ with a uniform constant involved in O(. Then by Theorem 3.3 and Proposition 3., we obtain EX j = + O Now we estimate E(X i X j, i j. We assume i<j. E(X i X j = 0 ( log = d,...,d j+ = d,...,d j ( (( j, E ( (( Xj 2 j/2 = 2 + O. ( ( d i+ log h i (d i + ( log ( d i+ log h i (d i + d j+ dλ h j (d j + ( d i+ log λ ( B(d,...,d j h i (d i + d j+ h j (d j + d j+ λ ( B(d,...,d j+ h j (d j + ( d j+ log Υ ( h j (d j +,d j+,y j, h j (d j + where the summation takes over all admissible vectors (d,d 2,...,d j+. By applying Lemma 2.2, the inner sum is bounded by a constant M. So, Combining (3.8 and (3.9, we get E(X i X j M E(X i. (3.9 (( n 2 E X i = O(n. (3.0 i= Thus {X j } satisfies all conditions in Lemma 3.8. This completes the proof. For R n, we also have the following weak law of large numbers. Theorem 3.0. Assume (H. Then the sequence n log n nj= R j (x converges in law to.

15 78 A.-H. Fan et al. / Journal of Number Theory 27 ( Proof. Fix n. For any k n, define U k (x = R k (x {Rk (x n log n}(x and V k (x = R k (x {Rk (x>n log n}(x. Set A n ={x (0, : R k = U k, k n}. Then By (3.4, we have { } n λ x (0, : R k (x n log n >ɛ k= { } n λ x A n : U k (x n log n >ɛ + λ ( A c n. k= λ { x (0, : V k (x 0 } = λ { x (0, : R k (x > n log n } thus λ(a c n n 2 k= n log n+ 0asn. The following is devoted to showing that { λ x (0, : n log n } n U k (x >ɛ 0. k= 2 n log n +, Using similar arguments in estimating E(X i X j in Theorem 3.9, and by Lemma 2.3 prepared before and Proposition 3., we have, for all k n, E ( U k (x = log n + o(log n, Therefore, by Chebyshev inequality, we get That is to say, lim n EU n (x n. { n λ x (0, : U k (x k= ( n Var U k (x = o ( n 2 (log n 2. (3. k= n E ( U k (x >ɛ k= n E ( U k (x } = o(. nk= E(U k (x nk= U k (x converges in probability to. Since (3. gives log n =, then we get lim nk= n n log n E(U k (x =. Thus P(A n 0 as 3.4. Central limit theorem for log R n (x In this subsection, we consider some central limit theorems related to R n (x = h n (d n (x+. We will show that Rn converges in law to the uniform distribution on [0, ] and log R n satisfies the central limit theorem (CLT. k= d n+(x

16 A.-H. Fan et al. / Journal of Number Theory 27 ( Proposition 3.. Assume (H. We have for any 0 <c<, lim λ{ x (0, : Rn <c } = c. (3.2 j Proof. It is easy to know that z j R n (x <z j + d j+. Then by Theorem 3.2 that d j + for almost all x (0, and by Theorem 3.3 that Z j U(0,, we get (3.2. Theorem 3.2. Assume (H. Then n nj= log R n converges in law to the normal distribution N(,. Proof. By the continuity theorem of characteristic functions [3, p. 9], we only need to show that the characteristic function of (log R + log R 2 + +log R n n n /2, denoted by ψ n (t, converges to e 2 t2. Let ϕ n (t = E ( e it(log R + +log R n ( t and ψ n (t = ϕ n n /2 e itn/2. Similar to estimate E(X i X j in Theorem 3.9, in the light of Lemma 2.4, we have ϕ n(t it ϕ t ( n (t n. 2 Thus ϕ n(t ( it n n j=0 t ( n j M t 2 for some absolute constant M. Set ϕ n (t = β n (t + D n (t t with D n (t M, where β(t = it.wehave ( ( ψ n (t = β n 3.5. Some special cases t n /2 ( t t + D n n /2 n /2 e itn/2 e 2 t2 as n. In this subsection, we give some applications of Theorems 3.9, 3.0 and 3.2. Example. Engel continued fraction expansion case: h j (n = n for all j and n. Then R n (x = d n+(x d n (x for each n. The results in Theorems 3.9, 3.0 and 3.2 are stated as follows. Corollary 3.3. (See []. For Engel continued fraction expansion, we have (i log d n+(x n converges in law to N(,.

17 80 A.-H. Fan et al. / Journal of Number Theory 27 ( (ii For almost all x (0,, lim n n log d n (x =. (iii nj= d j+ (x n log n d j (x converges in law to. Example 2. Sylvester continued fraction expansion case: h j (n = n(n for all j and d n. Then R n (x = n+ (x d n (x(d n (x + for each n. Corollary 3.4. For Sylvester continued fraction expansion, we have (i log d n+(x d (x dn(x n converges in law to N(,. d (ii For almost all x (0,, lim n n log n+ (x d (x d n (x =. (iii nj= converges in law to. n log n d j+ (x d j (x(d j (x + 4. The convergence speed In this section, we study the magnitude of x p n(x q n (x when {h j (n} j are polynomials of n, which will provide means to compare some known expressions covered by the Oppenheim continued fraction algorithm. At first, we give an estimation of the magnitude of x p n(x q n (x for the general case. Theorem 4.. Assume (H. Then for almost all x (0,, Proof. Similar to (2.6, since log x p n = ( + o( n log d j. (4. q n j= 2d n+ <x n+ d n+ and d n q n <q n 2d n q n,wehave nj= 2 n 2 R j nj= d x p n nj= R j nj=. (4.2 j d j By Theorems 3.2 and 3.9, respectively, we have q n lim n n n j= log d j = lim n log d n =+, n log R j = n + o(n j= λ-a.e. Thus n j= log R j = o( n j= log d j. As a consequence, we get the claim. Now we consider the special cases when h j (n are polynomials of n for all j. Corollary 4.2. Suppose for all j, h j (n = An + B with integers A and B satisfying An + B n for any n, then for almost all x (0,, lim n n 2 log x p n(x q n (x = ( + log A. (4.3 2

18 A.-H. Fan et al. / Journal of Number Theory 27 ( Proof. For this special case, an application of Theorem 3.9 yields lim n n log d n(x = + log A, λ-a.e. Definition 4.3. Let t>. We say that the function h j (n is of the order t if there exist constants 0 <K K 2 (independent of j such that K h j (n/n t K 2 for all large n. Theorem 4.4. Let h j (n be of order t>. Then for almost all x (0,, lim n t n log d n(x = G(x (4.4 where } + G(x = t {log d (x + t n log d n+(x dn t (x. (4.5 n= Proof. It suffices only to show that G is meaningful almost everywhere, since N n= t n log d n+(x dn t (x = t N log d N+ (x log d (x. By the definition of z n (x = x n+ (h n (d n (x +, and d n+ + <x n+ d n+, we get, when n is large enough, K z n (x d n+(x d n+ (x h n (d n (x + (d n (x t = h n (d n (x + (d n (x t 2K 2 z n (x. Recalling the distribution of z j in Theorem 3.3, we get E( log d n+(x d t n (x M. Moreover E ( log d (x = log d d (d + := M 2 <. d = These two integrations assert that 0 G(x dx < + which implies G(x is meaningful for almost all x (0,. Corollary 4.5. If h j (n is of order t>, then for almost all x (0,, where G(x is defined by (4.5. lim n t n log x p n(x q n (x = t G(x, (4.6 t

19 82 A.-H. Fan et al. / Journal of Number Theory 27 ( From Corollaries 4.2 and 4.5, we have the following corollary immediately. Corollary 4.6. The Lebesgue measure of those x (0, which are of same Engel and Sylvester continued fraction expansions is 0. Acknowledgments The authors thank the referees for valuable suggestions. This work was partially done during the second author s visit to the LAMFA, CNRS UMR 640, Amiens. He would like to thank the institution for warm hospitality. References [] R.L. Adler, L. Flatto, The backward continued fraction map and geodesic flow, Ergodic Theory Dynam. Systems 4 (4 ( [2] P. Erdös, A. Rényi, P. Szüsz, On the Engel s and Sylvester s series, Ann. Univ. Sci. Budapest. Etvs. Sect. Math. ( [3] J. Galambos, The ergodic properties of the denominators in the Oppenheim expansion of real numbers into infinite series of rationals, Quart. J. Math. Oxford Sec. Ser. 2 ( [4] J. Galambos, Further ergodic results on the Oppenheim series, Quart. J. Math. Oxford Sec. Ser. 25 ( [5] J. Galambos, On the speed of the convergence of the Oppenheim series, Acta Arith. 9 ( [6] J. Galambos, Representations of Real Numbers by Infinite Series, Lecture Notes in Math., vol. 502, Springer, 976. [7] Y. Hartono, C. Kraaikamp, F. Sweigher, Algebraic and ergodic properties of a new continued fraction algorithm with nondecreasing partial quotients, J. Théor. Nombres Bordeaux 4 ( [8] M. Iosifescu, C. Kraaikamp, Metrical Theory of Continued Fractions, Mathematics and Its Applications, vol. 547, Kluwer Academic Publishers, Dordrecht, [9] Lehner Joseph, Semiregular continued fractions whose partial denominators are or 2, in: Contemp. Math., vol. 69, 994, pp [0] C. Kraaikamp, A new class of continued fraction expansions, Acta Arith. 57 ( ( [] C. Kraaikamp, J. Wu, On a new continued fraction expansion with nondecreasing partial quotients, Monatsh. Math. 43 (4 ( [2] Y.Y. Liu, J. Wu, Hausdorff dimension in Engel expansions, Acta Arith. 99 ( [3] M. Loéve, Probability Theory, third ed., Van Nostrand, Princeton, NJ, 963. [4] A. Oppenheim, The representation of real numbers by infinite series of rationals, Acta Arith. 2 ( [5] A. Rényi, A new approach to the theory of Engel s series, Ann. Univ. Sci. Budapest. Etvs. Sect. Math. 5 ( [6] W. Sierpiński, Sur quelques algorithmes pour développer les nombres réel en séries, Oeuvres choisies, Tome I, PWN, Warszawa, 974, pp [7] F. Schweiger, Ergodische theorie der Engelchen und Sylvesterschen Reihen, Czechoslovak Math. J. 20 (95 ( [8] F. Schweiger, Ergodic Theory of Fibred Systems and Metric Number Theory, Clarendon Press, Oxford, 995. [9] F. Schweiger, Continued fraction with increasing digits, Österreich. Akad. Wiss. Math.-Nat. Kl. Sitzungsber. II 22 ( [20] M. Thaler, σ -Endliche invariante Masse für die Engelschen Reihen, Anz. Österreich. Akad. Wiss. Math.-Nat. Kl. 6 (2 ( [2] W. Vervaat, Success Epochs in Bernoulli Trials, Mathematical Center Tracts, vol. 42, Mathematisch Centrum, Amsterdam, 972. [22] J. Wu, A problem of Galambos on Engel expansions, Acta Arith. 92 ( [23] J. Wu, The Oppenheim series expansions and Hausdorff dimensions, Acta Arith. 07 (