HAUSDORFF DIMENSION OF CERTAIN SETS ARISING IN ENGEL CONTINUED FRACTIONS

. In the present paper, we are concerned with the Hausdorﬀ dimension of certain sets arising in Engel continued fractions. In particular, the Hausdorﬀ dimension of sets (cid:8) x ∈ [0 , 1) : b n ( x ) ≥ φ ( n ) i.m. n ∈ N (cid:9) and (cid:8) x ∈ [0 , 1) : b n ( x ) ≥ φ ( n ) , ∀ n ≥ 1 (cid:9) are completely determined, where i.m. means inﬁnitely many, { b n ( x ) } n ≥ 1 is the sequence of partial quotients of the Engel continued fraction expansion of x and φ is a positive function deﬁned on natural numbers.


1.
Introduction. Given a real number, there are various ways to represent it as an expansion of digits or partial quotients, such as continued fractions (see Khintchine [17]) and series expansions (see Galambos [9] and Schweiger [23]). One of the most well-known representation of real numbers is regular continued fractions (RCFs). The regular continued fraction expansion of a real number can be induced by the RCF-map (or Gauss transformation) T : [0, 1) → [0, 1) given by T (0) := 0 and T (x) = 1/x − 1/x , ∀x > 0, where x denotes the greatest integer not exceeding x. Indeed, putting a 1 (x) = 1/x and a n+1 (x) = a 1 (T n (x)) for any n ≥ 1, every real number x ∈ [0, 1) can be written uniquely as The form (1) is said to be the regular continued fraction (RCF) expansion of x and a n (x), n ∈ N are called the partial quotients of the RCF expansion of x. If there exists some n ∈ N such that T k (x) = 0 for all k ≥ n, we say that the RCF expansion of x is finite and denote (1) by [a 1 (x), a 2 (x), · · · , a n (x)]. Otherwise, it is said to be infinite and denote (1) by [a 1 (x), a 2 (x), · · · , a n (x), · · · ]. It is known that a real number has an infinite RCF expansion if and only if it is irrational. That is to say, there is a one-to-one correspondence between irrational numbers and the sequences of partial quotients. So it will help us to understand irrational numbers better by studying some properties of the corresponding partial quotients. A well-known result on partial quotients is the Borel-Bernstein theorem (see [17,Theorem 30]), which states that for almost all x ∈ [0, 1) in the sense of Lebesgue measure, a n (x) ≥ φ(n) holds for infinitely many n ∈ N or just for finitely many n ∈ N according as the series n≥1 1/φ(n) diverges or converges, where φ is a positive function defined on natural numbers. As a consequence of this result, many sets consisting of all real numbers whose partial quotients are subject to some kind of restrictions have null Lebesgue measure. In fractal geometry, Hausdorff dimension provides a very useful tool to measure such sets of Lebesgue measure zero and it is attained much attention in studying the exceptional sets arising in regular continued fractions. It is worth pointing out that the first published work in this region is due to Jarník [15], in which he investigated the set of real numbers whose partial quotients are bounded. In 1941, Good [10] gave a quite overall study of sets with some restrictions on partial quotients, including the set {x ∈ [0, 1) : a n (x) → ∞ as n → ∞}. For any positive function φ defined on natural numbers, he also attempted to investigate the set E(φ) = x ∈ [0, 1) : a n (x) ≥ φ(n) i.m. n ∈ N but did not give the exact value of its Hausdorff dimension, where i.m. means infinitely many. Furthermore, many authors tried to perfect Good's work on the Hausdorff dimension of E(φ), for instance, the Hausdorff dimension of the set {x ∈ [0, 1) : a n (x) ∈ B, ∀n ≥ 1 and a n (x) → ∞ as n → ∞}(B ⊆ N is infinite) are derived by combining the results of Hirst [13] and Wang and Wu [24], see also Cusick [2]. Later, Feng et al. [8] and Luczak [20] considered the Hausdorff dimension of the set E(φ) in the case of φ(n) = a b n with a, b > 1. At last, in 2008, Wang and Wu [25] completely solved the problem on the Hausdorff dimension of E(φ). Over the last twenty years, with the fast developing of dynamical systems, there is a close connection between representation of real numbers and dynamical systems. Many expansions of real numbers can be generated by some infinite iterated function system (iIFS, see [11,21,22]). In particular, regular continued fractions can be generated by the iIFS f n : [0, 1] → [0, 1] defined by The problem on the Hausdorff dimension of E(φ) has been well improved in the context of iIFSs (see [1,16,26]). It should be pointed out that regular continued fractions is a 2-decaying iIFS system in the context of Jordan and Rams [16]. See Cao et al. [1], Liao and Rams [19], Zhang and Cao [26] for more general results from regular continued fractions to d-decaying iIFS systems (d > 1).
In the present paper, we are interested in a variation of the regular continued fraction expansion, namely Engel continued fractions (ECFs). We emphasize that ECFs can not be generated by some infinite iterated function system and hence that the general results on the Hausdorff dimension of E(φ) in the case of [1] and [16] can not be applied to ECFs, which is the main motivation of this paper. In 2002, Hartono et al. [12] first introduced this new continued fraction algorithm with non-decreasing partial quotients. Let T E : [0, 1) −→ [0, 1) be the ECF-map given This means that the ECF-map T E (x) is in fact equal to the RCF-map T (x) normalized by 1/x . That is to say, we can obtain the ECF-map by shrinking each branch of the RCF-map according to a certain ratio (as shown in the above figure). The main common thing of these two maps is that they are both piecewise maps with infinitely many nonlinear branches. The difference of them is that all branches of the RCF-map are full, while, for the ECF-map, these branches are not full except for the first one on the right, which implies that partial quotients are non-decreasing (see below statement). Besides, it is well known that RCF-map is ergodic and has finite invariant measure equivalent to Lebesgue measure; however, Hartono et al. [12] showed that T E has no finite invariant measure equivalent to Lebesgue measure, but that it has infinitely many σ-finite, infinite invariant measures. Similar to regular continued fractions, every real number x ∈ [0, 1) can be represented in the following form The form (2) is said to be the ECF expansion of x denoted by [[b 1 (x), b 2 (x), · · · , b n (x), · · · ]] and b n (x), n ∈ N are called the partial quotients of the ECF expansion of x. The algorithm of ECFs producing non-decreasing partial quotients, has very different behaviors with respect to RCFs. For instance, Kraaikamp and Wu [18] proved a strong law of large numbers for log b n (x), i.e., holds for almost all x ∈ [0, 1) in the sense of Lebesgue measure. Moreover, they also showed that the set of real numbers in which such a strong law of large numbers does not hold, has full Hausdorff dimension. Furthermore, Fan et al. [4] established a central limit theorem for log b n (x). Following this line of research, Fang et al. [7] considered the large and moderate deviation principles for ECF expansions (see also [5,6]). For the Hausdorff dimension of some sets in ECFs, Zhong and Tang [27] considered a Hirst's problem in the context of ECFs and their results indicate that there is a difference between RCFs and ECFs in this problem. Recently, Hu et al. [14] studied the efficiency of approximating real numbers by their convergents of ECFs. In particular, they estimated the Hausdorff dimension of the set of points whose ECF-convergents are the best approximations infinitely often and obtained the Hausdorff dimensions of the related sets defined by some growth rates of partial quotients in ECF expansions (i.e., the Luczak's problem in the context of ECFs). We are interested in the Hausdorff dimension of E(φ) in the context of Engel continued fractions to significantly extend the results of Zhong and Tang [27] and Hu et al. [14]. More precisely, we would like to completely give the exact Hausdorff dimension of the sets where φ is a positive function defined on natural numbers. We also study the Hausdorff dimension of certain sets defined by the growth rate of partial quotients in ECFs and other related sets. The rest of this paper is organized as follows. In Section 2, we introduce some definitions and notations of Engel continued fractions. Section 3 is devoted to estimating the Hausdorff dimension of n with a, b > 1, which play an important role in studying the Hausdorff dimension of F (φ), F (φ) and other related sets. The exact formulas on the Hausdorff dimension of certain sets related to the growth rate of partial quotients are presented in Section 4. In particular, we completely determine the Hausdorff dimension of F (φ), F (φ) and other related sets. In Section 5, we consider the Hausdorff dimension of the set related to the ratio between two consecutive partial quotients, which shares a dichotomy law according to Borel-Bernstein type theorem. Throughout the paper, we use | · | to denote the diameter of a subset of [0,1), H s the s-dimensional Hausdorff measure and dim H the Hausdorff dimension.
2. Preliminaries. In this section, we recall some definitions and several arithmetic properties of Engel continued fractions. We first give an elementary arithmetic property of the Engel continued fraction expansion in representing real numbers, see Hartono et al. [12] (see also Fan et al. [4]).
The following proposition, due to Fan et al. [4], gives a characterization of all admissible sequences occurring in the ECF expansion.
the cylinder of order n of the ECF expansion.
In other words, I(b 1 , · · · , b n ) is the set of points beginning with (b 1 , · · · , b n ) in their ECF expansions. The following result gives the structure and the length of cylinders, see Hartono et al. [12] (see also Fan et al. [4]).
and hence that its length satisfies where the quantity Q n satisfies the recursive formula Q n = b n Q n−1 + b n−1 Q n−2 under the conventions Q −1 = 0 and Q 0 = 1.
By the recursive formula of Q n , we obtain that which is very useful in estimating the length of the cylinder in the next section. We use the notation λ to denote the Lebesgue measure on [0, 1) and treat partial quotients {b n } n≥1 as the random variables defined on probability space ) means the Borel σ-algebra on [0,1). We know that {b n } n≥1 does not form a homogeneous Markov chain (see Remark 5 of [7]) but has the following property, which is important in the metric theory of Engel continued fractions (see [4,7,18]). We also emphasize that such a property is not true for regular continued fractions.
and the conditional probabilities for all n ≥ 1.
3. Auxiliary results. Let a, b > 1. The main purpose of this section is to determine the Hausdorff dimension of which play an important role in the next section. Since E(a, b) ⊆ F (a, b), we first give a lower bound of dim H E(a, b) and an upper bound of dim H F (a, b). For the upper bound of dim H F (a, b), we point out that Zhong and Tang [27] in fact gave an estimation by using the ideas of Luczak [20] in the case of regular continued fractions (see also Hu et al. [14]).
Now it remains to calculate the lower bound of dim H E(a, b). To do this, we need the following lemma, which serve as an important tool to estimate the lower bound of the Hausdorff dimension of a fractal set (see Example 4.6 in [3]).
be a decreasing sequence of sets and E = n≥0 E n . We assume that each E n is a union of a finite number of disjoint closed intervals (called basic intervals of order n) and each basic interval in E n−1 contains m n intervals of E n which are separated by gaps of lengths at least ε n . If m n ≥ 2 and ε n−1 > ε n > 0, then By constructing such a subset of E(a, b) and using the result of Lemma 3.2, we determine the lower bound of dim H E(a, b).
with the convention D 0 = ∅. It should be noted that D 1 is not empty since σ 1 can at least take value ( a b + 1). Hence that D n is not empty for all n ≥ 1. For any (σ 1 , · · · , σ n ) ∈ D n and 1 ≤ k ≤ n−1, we know σ k+1 ≥ (k+1)a b k+1 > (k+1)a b k ≥ σ k and σ 1 ≥ 1. Hence (σ 1 , · · · , σ n ) is admissible for ECF expansions by Proposition 2. This is to say, all elements in D n are admissible for ECF expansions. For any (σ 1 , · · · , σ n ) ∈ D n , put where the union is taken over all σ n+1 satisfying (σ 1 , · · · , σ n , σ n+1 ) ∈ D n+1 and cl(A) denotes the closure of a set A. Note that J(σ 1 , · · · , σ n ) is a closed interval in [0, 1] and we call it the basic interval of order n. For all n ≥ 1, denote with the convention E 0 = [0, 1] and Then E is a subset of E(a, b). Next we will estimate the Hausdorff dimension of E.
To do this, we should first calculate the number of basic intervals of order n contained in the basic interval of order (n − 1) and the length of the gap between two of them. By the definitions of D n and J(σ 1 , · · · , σ n ), we know such a number is m n := (n + 1)a b n − na b n . Now it remains to estimate the length of the gap. For any two different blocks (σ 1 , · · · , σ n ) and (σ 1 , · · · , σ n ) in D n , we have that one of the following two intervals and σ n ≤σ n+1 <(n+1)a b n+1 is the gap between J(σ 1 , · · · , σ n ) and J(σ 1 , · · · , σ n ). We only calculate the length of the interval in (7) since the calculation for the interval in (8) is similar. Let η = a b(b−1) > 1. Then we have that a b n+1 ≥ ηa b n holds for all n ≥ 1. If σ n ≤ (n + 1)a b n , then it follows that (n + 1)a b n+1 ≥ ησ n for all n ≥ 1. By (4) and (5), we deduce that |I(σ 1 , · · · , σ n , σ n+1 )| ≥ σn≤σn+1≤ησn |I(σ 1 , · · · , σ n , σ n+1 )| To summarize, we constructed a subset of E(a, b): E = n≥0 E n , where E 0 = [0, 1] and E n is defined in (6) which is a union of a finite number of disjoint basic intervals of order n. Moreover, each basic intervals of order (n − 1) contains m n basic intervals of order n and the length of the gap between two of them is at least ε n . Note that m n ≥ 2 and ε n > ε n+1 > 0 for all n ≥ 1, by Lemma 3.2, we obtain that .

LULU FANG AND MIN WU
For a, b > 1, let F (a, b), in view of Lemmas 3.1 and 3.3, we have Being similar to the proof of Lemma 3.3, we can obtain the following proposition, which shows that the set of x's such that b n (x) tends to infinity with exponential rates as n goes to infinity has full Hausdorff dimension.
Proposition 6. For any α > 0, then Proof. The proof is very similar to the proof of Lemma 3.3, so we just list the key steps. Let α > 0. The symbolic space D n can be defined as It is not difficult to show that all elements in D n are admissible for ECF expansions. For any (σ 1 , · · · , σ n ) ∈ D n , the basic interval of order n is defined as J(σ 1 , · · · , σ n ) = σn+1 cl(I(σ 1 , · · · , σ n , σ n+1 )).
As a consequence of Proposition 6, we have the following two corollaries.
Then L is an infinite subset of N . Therefore, It follows from Corollary 2 that F (φ) has full Hausdorff dimension.
Then L is an infinite subset of N. So, In view of Proposition 5, we deduce that In this case, for any B > 1, there exists N := N (B) > 0 such that for all n ≥ N , we have φ(n) ≥ e B n . Then, Similarly, we have the following result.
To prove Theorem 4.2, we need the following lemma.
In view of the countable stability of Hausdorff dimension and invariant property under bi-Lipschitz map (see [3]), it suffices to show that f B1,··· ,B N is a bi-Lipschitz map. In fact, for any x, y ∈ F N (φ), we have where P N and Q N satisfy the recursive formula: with the convention P 0 = 0, P 1 = 1 and Q 0 = 1, Q 1 = B 1 . Hence that Therefore, That is, f B1,··· ,B N is a bi-Lipschitz map.
Now we are ready to prove Theorem 4.2.
Proof of Theorem 4.2.
(2) 1 < β < ∞. Since log β = lim sup n→∞ log log φ(n) n , for any 0 < ε < β − 1, we have φ(n) ≥ e (β−ε) n holds for infinitely many n ∈ N. Also there exists N := N (ε) > 0 such that for all n > N , we have φ(n) ≤ e (β+ε) n . So, Combing these with Lemma 3.1, Proposition 5 and Lemma 4.3, we conclude that holds for infinitely many n ∈ N. Then We further investigate this topic by considering a more fast growth speed of log b n , which indicates that the corresponding Hausdorff dimension will decay with an inverse function rate when the growth speed of log b n is exponential.
Proof. On the one hand, for any α > 0, it is clear that E(a, b) is a subset of the desired set, which implies that by Lemma 5. On the other hand, let x ∈ [0, 1) satisfy For any 0 < ε < α, there exists N := N (x, ε) > 0 such that for all n ≥ N , we have b n (x) ≥ e (α−ε)b n . So, Applying Lemma 3.1 to a = e α−ε > 1, we deduce that The proof is completed.
holds for almost all x ∈ [0, 1) in the sense of Lebesgue measure. The following result will study the Hausdorff dimension of the set of points in which such a limit attains any other values.

5.
Ratio between two consecutive partial quotients. This section is devoted to dealing with the Hausdorff dimension of certain sets related to the ratio between two consecutive partial quotients. Such a ratio can be viewed as an indirect way to consider the growth rate of partial quotients. For any x ∈ [0, 1) and n ≥ 1, let Let φ be a positive function defined on natural numbers and R(φ) = x ∈ [0, 1) : R n (x) ≥ φ(n) i.m. n . For the size of R(φ), Fan et al. [4] gave a zero-one law for the Lebesgue measure of R(φ).
We will give a complete description of R(φ) from the viewpoint of fractal dimensions. To do this, let for a, b > 1. We first give the exact Hausdorff dimensions of R(a, b) and R(a, b).
Proof. Let a, b > 1. On the one hand, we know by the definition of R n (x) and hence that which implies that dim H R(a, b) ≤ 1/b by Proposition 5. On the other hand, for a, b > 1, let c > 0 satisfy the equation c b−1 = a, then c > 1. We claim that This is because that if nc b n ≤ b n (x) ≤ (n + 1)c b n holds for some x ∈ [0, 1) and for all n ≥ 1, we have Thus, it follows from Proposition 5 that dim H R(a, b) ≥ 1/b. Since R(a, b) ⊆ R(a, b), we conclude that dim H R(a, b) = dim H R(a, b) = 1/b.
The proofs of the following two theorems are very similar to the proof Theorems 4.1 and 4.2, and are left to the reader. (1) If β = 1, then dim H R(φ) = 1. (1) If β = 1, then dim H R(φ) = 1.
For a further investigation on the ratio R n , we consider L n (x) = max{R 1 (x), · · · , R n (x)} for x ∈ [0, 1) and n ≥ 1. In [4], Fan et al. proved an iterated logarithm type theorem for L n . We emphasize that the 0-1 law is also true for L n as a consequence of Theorem 5.1. has Lebesgue measure zero or one according as the series n≥1 1/φ(n) converges or diverges.
Proof. Let φ be a non-decreasing and positive function defined on natural numbers. If sup n≥1 φ(n) < ∞, then the series n≥1 1/φ(n) diverges and hence that for almost all x ∈ [0, 1), R n (x) ≥ φ(n) holds for infinitely many n ∈ N by Theorem 5.1. Note that L n (x) ≥ R n (x), so L n (x) ≥ φ(n) holds infinitely often for almost all x ∈ [0, 1). Now let φ(n) tend to ∞ as n goes to ∞. In this case, we claim that L(φ) = R(φ).
On the one hand, since L n (x) ≥ R n (x) for any x ∈ [0, 1) and n ≥ 1, we have R(φ) ⊆ L(φ). On the other hand, for any x ∈ [0, 1), if L n (x) ≥ φ(n) holds for infinitely many n ∈ N, then there exists a subsequence {n k } k≥1 (depending on x) such that L n k (x) ≥ φ(n k ) for all k ≥ 1. For all k ≥ 1, by the definition of L n k (x), there is m k ∈ {1, · · · , n k } such that R m k (x) = L n k (x) ≥ φ(n k ). Since φ is non-decreasing, we obtain R m k (x) ≥ φ(m k ). Note that φ(n) → ∞ as n → ∞, it is clear that for all k ≥ 1, we can not choose the same m k . Therefore, L(φ) ⊆ R(φ).
From the proof of Proposition 7, we can see that if sup n≥1 φ(n) < ∞, then R n (x) ≥ φ(n) and L n (x) ≥ φ(n) both hold infinitely often for almost all x ∈ [0, 1). Of course, R(φ) and L(φ) have full Hausdorff dimension. If φ(n) tends to ∞ as n goes to ∞, then R(φ) = L(φ). Therefore, we always have that R(φ) and L(φ) share the same Hausdorff dimension when φ is a non-decreasing and positive function defined on natural numbers. However, the following result shows that the assumption of non-decreasing property on function φ can be relaxed.