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arXiv:1608.04326v2 [math.NT] 27 Aug 2016 A REMARK ON THE EXTREME VALUE THEORY FOR CONTINUED FRACTIONS LULU FANG AND KUNKUN SONG Abstract. Let x be a irrational number in the unit interval and denote by its continued fraction expansion [a1 (x), a2 (x), · · · , an (x), · · · ]. For any n ≥ 1, write Tn (x) = max1≤k≤n {ak (x)}. We are interested in the Hausdorff dimension of the fractal set Tn (x) Eφ = x ∈ (0, 1) : lim =1 , n→∞ φ(n) where φ is a positive function defined on N with φ(n) → ∞ as n → ∞. Some partial results have been obtained by Wu and Xu, Liao and Rams, and Ma. In the present paper, we further study this topic when φ(n) tends to infinity with a doubly exponential rate as n goes to infinity. 1. Introduction Every irrational number x in the unit interval has a unique continued fraction expansion of the form 1 x= a1 (x) + , 1 . a2 (x) + . . + (1.1) 1 . an (x) + . . where an (x) are positive integers and are called the partial quotients of the continued fraction expansion of x (n ∈ N). Sometimes we write the representation (1.1) as [a1 (x), a2 (x), · · · , an (x), · · · ]. For more details about continued fractions, we refer the reader to a monograph of Khintchine [11]. Let x ∈ (0, 1) be an irrational number. For any n ≥ 1, we define Tn (x) = max{ak (x) : 1 ≤ k ≤ n}, i.e., the largest one in the block of the first n partial quotients of the continued fraction expansion of x. Extreme value theory in probability theory is concerned with the limit distribution laws for the maximum of a sequence of random variables (see [12]). Galambos [5] first considered the extreme value theory for continued fractions and obtained that y Tn (x) lim µ x ∈ (0, 1) : < = e−1/y n→∞ n log 2 2010 Mathematics Subject Classification. Primary 11K50, 28A80; Secondary 60G70. Key words and phrases. Continued fractions, Extreme value theory, Doubly exponential rate, Hausdorff dimension. 1 2 LULU FANG AND KUNKUN SONG for any y > 0, where µ is equivalent to the Lebesgue measure, namely Gauss measure given by Z 1 1 µ(A) = dx log 2 A 1 + x for any Borel set A ⊆ (0, 1). Later, he also gave an iterated logarithm type theorem for Tn (x) in [6], that is, for µ-almost all x ∈ (0, 1), log Tn (x) − log n =1 log log n n→∞ As a consequence, we know that lim sup and lim inf n→∞ log Tn (x) − log n = 0. log log n log Tn (x) =1 log n holds for µ-almost all x ∈ (0, 1). Furthermore, Philipp [16] solved a conjecture of Erdős for Tn (x) and obtained its order of magnitude. In fact, lim n→∞ 1 Tn (x) log log n = . n log 2 holds for µ-almost all x ∈ (0, 1). Later, Okano [15] constructed some explicit real numbers that satisfied this liminf. However, a natural question arises: what is the exact growth rate of Tn or whether there exists a normalizing sequence {bn }n≥1 such that Tn (x)/bn converges to a positive and finite constant for µ-almost all x ∈ (0, 1). Unfortunately, there is a negative answer for this question. That is to say, there is no such a normalizing non-decreasing sequence {bn }n≥1 so that Tn (x)/bn converges to a positive and finite constant for µ-almost all x ∈ (0, 1). More precisely, lim inf n→∞ Tn (x) =0 bn Tn (x) lim sup = +∞ bn n→∞ P holds for µ-almost all x ∈ (0, 1) according to n≥1 1/bn converges or diverges. In 1935, Khintchine [10] proved that Sn (x)/(n log n) converges in measure to the constant 1/(log 2) and Philipp [17] remarked that this result cannot hold for µalmost all x ∈ (0, 1), where Sn (x) = a1 (x) + · · · + an (x). That is to say, the strong law of large numbers for Sn fails. However, Diamond and Vaaler [3] showed that the maximum Tn (x) should be responsible for the failure of the strong law of large numbers, i.e., 1 Sn (x) − Tn (x) = lim n→∞ n log n log 2 for µ-almost all x ∈ (0, 1). These results indicate that the maximum Tn (x) play an important role in metric theory of continued fractions. For more metric results on extreme value theory for continued fractions, we refer the reader to Barbolosi [1], Bazarova et al. [2], Iosifescu and Kraaikamp [7] and Kesseböhmer and Slassi [8, 9]. The following will study the maximum Tn (x) from the viewpoint of the fractal dimension. More precisely, we are interested in the Hausdorff dimension of the fractal set Tn (x) =1 , Eφ = x ∈ (0, 1) : lim n→∞ φ(n) where φ is a positive function defined on N with φ(n) → ∞ as n → ∞. Wu and Xu [19] have obtained some partial results on this topic and they pointed out that Eφ has full Hausdorff dimension if φ(n) tends to infinity with a polynomial rate as n goes to infinity. lim sup n→∞ or A REMARK ON THE EXTREME VALUE THEORY FOR CONTINUED FRACTIONS 3 Theorem 1.1 ([19]). Assume that φ is a positive function defined on N satisfying φ(n) → ∞ as n → ∞ and log φ(n) lim < ∞. n→∞ log n Then Tn (x) = 1 = 1. dimH x ∈ (0, 1) : lim n→∞ φ(n) Liao and Rams [13] considered the Hausdorff dimension of Eφ when φ(n) tends to infinity with a single exponential rate. Also, they showed that there is a jump α of the Hausdorff dimensions from 1 to 1/2 on the class φ(n) = en at α = 1/2. Theorem 1.2 ([13]). For any β > 0, ( 1, Tn (x) =β = dimH x ∈ (0, 1) : lim α n→∞ en 1/2, α ∈ (0, 1/2); α ∈ (1/2, +∞). However, they don’t know what will happen at the critical point α = 1/2. Recently, Ma [14] solved this left unknown problem and proved that its Hausdorff dimension is 1/2 when α = 1/2. In the present paper, we will further investigate the Hausdorff dimension of Eφ when φ(n) tends to infinity with a doubly exponential rate. Moreover, we will see that the Hausdorff dimension of Eφ will decay to zero if the speed of φ(n) is growing faster and faster, which can be treated as a supplement to Wu and Xu [19], Liao and Rams [13], and Ma [14] in this topic. Theorem 1.3. Let b, c > 1 be real numbers. Then for any β > 0, α ∈ (0, 1); 1/2, Tn (x) dimH x ∈ (0, 1) : lim bnα = β = 1/(b + 1), α = 1; n→∞ c 0, α ∈ (1, +∞). For more results about the Hausdorff dimensions of fractal sets related to Tn , see Zhang [21], and Zhang and Lü [22]. The following figure is an illustration of the Haudorff dimension of Eφ for different φ. Figure 1. dimH Eφ for different φ 4 LULU FANG AND KUNKUN SONG 2. Preliminaries This section is devoted to recalling some definitions and basic properties of continued fractions. Let x ∈ (0, 1) be a irrational number and its continued fraction expansion x = [a1 (x), a2 (x), · · · , an (x), · · · ]. For any n ≥ 1, we denote by pn (x) := [a1 (x), a2 (x), · · · , an (x)] qn (x) the n-th convergent of the continued fraction expansion of x, where pn (x) and qn (x) are relatively prime. With the conventions p−1 = 1, q−1 = 0, p0 = 0, q0 = 1, the quantities pn (x) and qn (x) satisfy the following recursive formula: pn (x) = an (x)pn−1 (x) + pn−2 (x) and qn (x) = an (x)qn−1 (x) + qn−2 (x), (2.1) which implies that a1 (x)a2 (x) · · · an (x) ≤ qn (x) ≤ (a1 (x) + 1)(a2 (x) + 1) · · · (an (x) + 1). (2.2) Definition 2.1. For any n ≥ 1 and a1 , a2 , · · · , an ∈ N, we call I(a1 , · · · , an ) := {x ∈ (0, 1) : a1 (x) = a1 , · · · , an (x) = an } the n-th order cylinder of continued fraction expansion. In other words, I(a1 , · · · , an ) is the set of points beginning with (a1 , a2 , · · · , an ) in their continued fraction expansions. Proposition 2.2. Let n ≥ 1 and a1 , a2 , · · · , an ∈ N. Then I(a1 , · · · , an ) is an interval with two endpoints pn qn and pn + pn−1 . qn + qn−1 More precisely, pn /qn is the left endpoint if n is even; otherwise it is the right endpoint. Moreover, the length of I(a1 , · · · , an ) satisfies |I(a1 , · · · , an )| = 1 , qn (qn + qn−1 ) (2.3) where pn and qn satisfy the recursive formula (2.1). 3. Proof of Theorem 1.3 In this section, we will give the proof of Theorem 1.3 which is inspired by Xu [20]. More precisely, we will determine the Hausdorff dimension of the fractal set Tn (x) E(b, c, α, β) = x ∈ (0, 1) : lim bnα = β n→∞ c for b, c > 1 and α, β > 0. The proof is divided into two parts: upper bound and lower bound for dimH E(b, c, α, β). A REMARK ON THE EXTREME VALUE THEORY FOR CONTINUED FRACTIONS 5 3.1. Upper bound. Let b, c > 1 be real numbers. For any α > 0, we define n o nα E ∗ (b, c, α) := x ∈ (0, 1) : an (x) ≥ cb i.o. , where i.o. denotes infinitely often. The following result is a special case of Theorem 4.2 in Wang and Wu [18]. Lemma 3.1. ([18, Theorem 4.2]) Let b, c > 1 be 1/2, dimH E ∗ (b, c, α) = 1/(b + 1), 0, a real number. Then α ∈ (0, 1); α = 1; α ∈ (1, +∞). Now we show that E(b, c, α, β) is a subset of F (d, c, α) with b > d > 1. Lemma 3.2. Let b, c > 1 and α, β > 0. Then for any b > d > 1, we have E(b, c, α, β) ⊆ E ∗ (d, c, α). α (n+1)α α nα −b Proof. Since b(n+1) − bn → +∞ for any α > 0, we know that cb ≥ c for sufficiently large n. Choose 0 < δ < β enough small such that (β − δ) · c > β + δ. For any x ∈ E(b, c, α, β), we know that lim n→∞ Tn (x) = β. α cbn Note that (β − δ)cb (n+1)α − (β + δ)cb nα α cd(n+1) (n+1)α α cb bn = d(n+1)α · β − δ − (β + δ) · (n+1)α b c α α 1 b(n+1) −d(n+1) ≥c · β − δ − (β + δ) · → ∞, c then there exists N0 > 0 (depending on δ) such that for any n ≥ N0 , we have (β − δ)cb nα ≤ Tn (x) ≤ (β + δ)cb nα and (β − δ)cb (n+1)α nα − (β + δ)cb ≥ cd (n+1)α . Thus, we actually deduce that an+1 (x) ≥ Tn+1 (x) − Tn (x) ≥ (β − δ)cb (n+1)α holds for any n ≥ N0 . So we have x ∈ F (d, c, α). − (β + δ)cb nα (n+1)α ≥ cd It follows from Lemma 3.1 that dimH E(b, c, α, β) ≤ 1/2 for 0 < α < 1 and dimH E(b, c, α, β) = 0 for α > 1. When α = 1, by Lemmas 3.1 and 3.2, we know that dimH E(b, c, α, β) ≤ dimH F (d, c, α) ≤ 1/(d + 1) for any b > d > 1. So dimH E(b, c, α, β) ≤ 1/(b + 1) by letting d → b+ . Therefore, we eventually obtain that α ∈ (0, 1); 1/2, dimH E(b, c, α, β) ≤ 1/(b + 1), α = 1; 0, α ∈ (1, +∞). 6 LULU FANG AND KUNKUN SONG 3.2. Lower bound. For any n ∈ N, we define α α n n 1 1 f (n) = β − cb and g(n) = β + cb . n n Then f (n) → ∞ and g(n) → ∞ as n → ∞. So we can choose N > 0 sufficiently large such that nα cb f (n) ≥ 2, g(n) is non-decreasing and ≥2 n for all n ≥ N . Let FN (b, c, α, β) = x ∈ (0, 1) : f (n) ≤ an (x) ≤ g(n) for all n ≥ N . The following lemma states that FN (b, c, α, β) is a subset of E(b, c, α, β). Lemma 3.3. FN (b, c, α, β) ⊆ E(b, c, α, β). Proof. For any x ∈ FN (b, c, α, β), we have f (n) ≤ an (x) ≤ g(n) (3.1) for all n ≥ N . Note that g(n) is non-decreasing for all n ≥ N , we know that Tn (x) ≤ max TN −1 (x), g(n) ≤ TN −1 (x) + g(n). Combing this with (3.1), we deduce that f (n) ≤ an (x) ≤ Tn (x) ≤ TN −1 (x) + g(n) for any n ≥ N . Therefore, Tn (x) =β α cbn and hence x ∈ E(b, c, α, β). That is to say, FN (b, c, α, β) ⊆ E(b, c, α, β). lim n→∞ Next we estimate the lower bound for the Hausdorff dimenson of FN (b, c, α, β). To do this, we need the following lemma, which provides a method to obtain a lower bound Hausdorff dimension of a fractal set (see [4, Example 4.6]). T Lemma 3.4. Let E = n≥0 En , where [0, 1] = E0 ⊃ E1 ⊃ · · · is a decreasing sequence of subsets in [0, 1] and En is a union of a finite number of disjoint closed intervals (called n-th level intervals) such that each interval in En−1 contains at least mn intervals of En which are separated by gaps of lengths at least εn . If mn ≥ 2 and εn−1 > εn > 0, then dimH E ≥ lim inf n→∞ log(m1 m2 · · · mn−1 ) . − log(mn εn ) Proof. Suppose the liminf is positive, otherwise the result is obvious. We may assume that each interval in En−1 contains exactly mn intervals of En since we can remove some excess intervals to get smaller sets En0 and E 0 . Thus the gaps of lengths between different intervals in En are not changed and hence we just need to deal with these new smaller sets. Now we define a mass distribution µ on E by assigning a mass of (m1 · · · mn )−1 to each of (m1 · · · mn ) n-th level intervals in En . Next we will check the conditions of the classical Mass distribution principle. Let U be an interval of length |U | satisfying 0 < |U | < ε1 . Then there exists a integer n such that εn ≤ |U | < εn−1 . On the one hand, U can intersect at most one (n − 1)-th level interval since the gap of length between different intervals in En−1 is at least εn−1 and hence U can intersect at most mn n-th level intervals; on the A REMARK ON THE EXTREME VALUE THEORY FOR CONTINUED FRACTIONS 7 other hand, since |U | ≥ εn and the gap of length between different intervals in En is at least εn , we know U can intersect at most (|U |/εn + 1) ≤ 2|U |/εn n-th level intervals. Note that each n-th level interval has mass (m1 · · · mn )−1 , so µ(U ) ≤ min{mn , 2|U |/εn } · (m1 · · · mn )−1 ≤ m1−s · (2|U |/εn )s · (m1 · · · mn )−1 n for any 0 ≤ s ≤ 1 and hence 2s µ(U ) ≤ . s |U | (m1 · · · mn−1 )msn εsn Let s > lim inf log(m1 · · · mn−1 )/(− log(mn εn )). Then (m1 · · · mn−1 )msn εsn > 1 for n→∞ sufficiently large n. Thus µ(U ) ≤ C · |U |s , where C > 0 is an absolute constant. By the classical Mass distribution principle (see [4, Chapter 4]), we have dimH E ≥ s. This completes the proof. Lemma 3.5. 1/2, dimH FN (b, c, α, β) ≥ 1/(b + 1), 0, α ∈ (0, 1); α = 1; α ∈ (1, +∞). Proof. For any n ≥ 1, we define n kα Dn = (σ1 , · · · , σn ) ∈ Nn : 1 ≤ σk ≤cb /k + 1 for all 1 ≤ k < N and f (k) ≤ σk ≤ g(k) for all N ≤ k ≤ n o and D= [ Dn n≥0 with the convention D0 := ∅. For any n ≥ 1 and (σ1 , · · · , σn ) ∈ Dn , we denote [ J(σ1 , · · · , σn ) = clI(σ1 , · · · , σn , σn+1 ) σn+1 and call it the n-th level interval, where the union is taken over all σn+1 such that (σ1 , · · · , σn , σn+1 ) ∈ Dn+1 , cl denotes the closure of a set and I(σ1 , · · · , σn , σn+1 ) is the (n + 1)-th cylinder for continued fractions. Let [ E0 = [0, 1], En = J(σ1 , · · · , σn ) (σ1 ,··· ,σn )∈Dn T for any n ≥ 1 and E := n≥0 En . Then E is a Cantor-like subset of FN (b, c, α, β). It follows from the construction of E that each element in En−1 contains some number of the n-th level intervals in En . We denote such a number by Mn . If 1 ≤ n < N , by the definition of Dn , we have $ nα % cb Mn = + 1, n where bxc denotes the greatest integer not exceeding x. When n ≥ N , we get $ nα % $ nα % 2cb cb Mn ≥ ≥ + 1. n n 8 LULU FANG AND KUNKUN SONG j nα k So we obtain Mn ≥ mn := cb /n + 1. Next we estimate the gaps between the same order level intervals. For any n ≥ N and two distinct level intervals J(τ1 , · · · , τn ) and J(σ1 , · · · , σn ) of En , we assume that J(τ1 , · · · , τn ) locates in the left of J(σ1 , · · · , σn ) without loss of generality. By Proposition 2.2, we know the level intervals J(τ1 , · · · , τn ) and J(σ1 , · · · , σn ) are separated by the (n + 1)-th cylinder I(τ1 , · · · , τn , 1) or I(σ1 , · · · , σn , 1) according to n is even or odd. In fact, if n is odd, note that J(τ1 , · · · , τn ) is a union of a finite number of the closure of (n + 1)-th order cylinders like I(τ1 , · · · , τn , j) with 2 ≤ f (n + 1) ≤ j ≤ g(n + 1) and these cylinders run from right to left, and so is J(σ1 , · · · , σn ), therefore J(τ1 , · · · , τn ) and J(σ1 , · · · , σn ) are separated by I(τ1 , · · · , τn , 1) in this case. When n is even, they are separated by I(σ1 , · · · , σn , 1). Thus the gap is at least |I(τ1 , · · · , τn , 1)| or |I(σ1 , · · · , σn , 1)|, where | · | denotes the length of a interval. In view of (2.2) and (2.3), we deduce that !−2 n Y 1 1 ≥ · (τk + 1) |I(τ1 , · · · , τn , 1)| ≥ 2 2qn+1 8 k=1 ! n !−2 kα N −1 Y Y 1 1 bkα cb = 2n+3 · +1 β+ c 2 k k k=1 k=N := εn . Similarly, we can also obtain |I(σ1 , · · · , σn , 1)| ≥ εn . It is easy to check that εn > εn+1 > 0 for sufficiently large n and εn → 0 as n → ∞. These imply that the gaps between any two n-th level intervals are at least εn . By Lemma 3.4, we have dimH E ≥ lim inf n→∞ log(m1 m2 · · · mn ) − log(mn+1 εn+1 ) α ≥ lim inf n→∞ α α b1 + b2 + · · · + bn 2(b1α + · · · + b(n+1)α ) − b(n+1)α α = lim inf n→∞ α 2(b1α α b1 + · · · + bn . + · · · + bnα ) + b(n+1)α α α When 0 < α < 1, then b(n+1) /(b1 + · · · + bn ) → 0 as n → ∞ and hence dimH E ≥ 1/2. If α = 1, we know bn+1 /(b + · · · + bn ) → b − 1 as n → ∞ and hence α α α dimH E ≥ 1/(b + 1). When α > 1, we have b(n+1) /(b1 + · · · + bn ) → +∞ as n → ∞ and hence dimH E ≥ 0. Since E is a subset of FN (b, c, α, β), we complete the proof. Combing Lemmas 3.3 and 3.5, we get α ∈ (0, 1); 1/2, dimH E(b, c, α, β) ≥ dimH FN (b, c, α, β) ≥ 1/(b + 1), α = 1; 0, α ∈ (1, +∞). The results on the upper and lower bounds imply the proof of Theorem 1.3. A REMARK ON THE EXTREME VALUE THEORY FOR CONTINUED FRACTIONS 9 Remark 3.6. In fact, from the proof of the lower bound, we can obtain that dimH Eφ ≥ lim inf n→∞ log φ(1) + · · · + log φ(n) . 2(log φ(1) + · · · + log φ(n)) + log φ(n + 1) Therefore, we always have dimH Eφ ≥ 1/2 when φ(n) tends to infinity with single exponential rates since lim sup n→∞ (n + 1)α =0 1α + 2α + · · · + nα for any α > 0. And we also get dimH E(b, c, α, β) ≥ 1/2 when 0 < α < 1 and dimH E(b, c, α, β) ≥ 1/(b + 1) if α = 1 since α lim sup n→∞ b1α b(n+1) =0 + b2α + · · · + bnα for 0 < α < 1 and it is (b − 1) when α = 1. Moreover, it also indicates that the Hausdorff dimension of Eφ is just related to the second base (i.e., b) in the doubly exponential rate when α = 1. References [1] D. Barbolosi, Sur l’ordre de grandeur des quotients partiels du développement en fractions continues régulières (French), Monatsh. Math. 128 (1999), no. 3, 189–200. [2] A. Bazarova, I. Berkes and L. Horváth, On the extremal theory of continued fractions, J. Theoret. Probab. 29 (2016), no. 1, 248–266. [3] H. Diamond and J. Vaaler, Estimates for partial sums of continued fraction partial quotients, Pacific J. Math. 122 (1986), no. 1, 73–82. [4] K. Falconer, Fractal Geometry: Mathematical Foundations and Applications, John Wiley & Sons, Ltd., Chichester, 1990. [5] J. Galambos, The distribution of the largest coefficient in continued fraction expansions, Quart. J. Math. Oxford Ser. (2) 23 (1972), 147–151. [6] J. Galambos, An iterated logarithm type theorem for the largest coefficient in continued fractions, Acta Arith. 25 (1973/74), 359–364. [7] M. Iosifescu and C. Kraaikamp, Metrical Theory of Continued Fractions. Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, 2002. [8] M. Kesseböhmer and M. Slassi, Large deviation asymptotics for continued fraction expansions, Stoch. Dyn. 8 (2008), no. 1, 103–113. [9] M. Kesseböhmer and M. Slassi, A distributional limit law for the continued fraction digit sum, Math. Nachr. 281 (2008), no. 9, 1294–1306. [10] A. Khintchine, Metrische Kettenbruchprobleme, Compositio Math. 1 (1935), 361–382. [11] A. Khintchine, Continued Fractions, The University of Chicago Press, Chicago, 1964. [12] M. Leadbetter, G. Lindgren and H. Rootzén, Extremes and related properties of random sequences and processes, Springer Series in Statistics. Springer-Verlag, New York-Berlin, 1983. [13] L.-M. Liao and M. Rams, Subexponentially increasing sums of partial quotients in continued fraction expansions, Math. Proc. Cambridge Philos. Soc. 160 (2016), no. 3, 401–412. [14] L.-G. Ma, A remark on Liao and Rams’ result on distribution of the leading partial quotient 1/2 [15] [16] [17] [18] [19] with growing speed en in continued fractions, arXiv:1606.03344. T. Okano, Explicit continued fractions with expected partial quotient growth, Proc. Amer. Math. Soc. 130 (2002), no. 6, 1603–1605. W. Philipp, A conjecture of Erdős on continued fractions, Acta Arith. 28 (1975/76), no. 4, 379–386. W. Philipp, Limit theorems for sums of partial quotients of continued fractions, Monatsh. Math., 105 (1988), no. 3, 195–206. B.-W. Wang and J. Wu, Hausdorff dimension of certain sets arising in continued fraction expansions, Adv. Math. 218 (2008), no. 5, 1319–1339. J. Wu and J. Xu, The distribution of the largest digit in continued fraction expansions, Math. Proc. Cambridge Philos. Soc. 146 (2009), no. 1, 207–212. 10 LULU FANG AND KUNKUN SONG [20] J. Xu, On sums of partial quotients in continued fraction expansions, Nonlinearity 21 (2008), no. 9, 2113–2120. [21] Z.-L. Zhang, The relative growth rate of the largest digit in continued fraction expansions, Lith. Math. J. 56 (2016), no. 1, 133–141. [22] Z.-L. Zhang and M.-Y. Lü, The relative growth rate of the largest partial quotient to the sum of partial quotients in continued fraction expansions, J. Number Theory 163 (2016), 482–492. School of Mathematics, South China University of Technology, Guangzhou 510640, P.R. China E-mail address: [email protected] School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, P.R. China E-mail address: [email protected]