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Hausdorff dimensions of some exceptional sets in Engel expansions
Accepted Manuscript Hausdorff dimensions of some exceptional sets in Engel expansions
Meiying Lü, Jia Liu
PII: DOI: Reference:
S0022-314X(17)30356-6 https://doi.org/10.1016/j.jnt.2017.09.015 YJNTH 5885
To appear in:
Journal of Number Theory
Received date: Revised date: Accepted date:
18 April 2017 18 September 2017 24 September 2017
Please cite this article in press as: M. Lü, J. Liu, Hausdorff dimensions of some exceptional sets in Engel expansions, J. Number Theory (2018), https://doi.org/10.1016/j.jnt.2017.09.015
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
HAUSDORFF DIMENSIONS OF SOME EXCEPTIONAL SETS IN ENGEL EXPANSIONS ¨ 1,∗ , JIA LIU 2 MEIYING LU Abstract. Given any real number x ∈ (0, 1], denote its Engel expansion by ∞ 1 n=1 d1 (x)···dn (x) , where {d j (x), j ≥ 1} is a sequence of positive integers satisfying d1 (x) ≥ 2 and d j+1 (x) ≥ d j (x)( j ≥ 1). Suppose φ : N → R+ is a function satisfying φ(n + 1) − φ(n) → ∞ as n → ∞. In this paper, we consider the set log dn (x) =1 , E(φ) = x ∈ (0, 1] : lim n→∞ φ(n) and we quantify the size of E(φ) in the sense of Hausdorff dimension. As appli cations, we get the Hausdorff dimensions of the sets x ∈ (0, 1] : lim logndβn (x) = γ n→∞ and x ∈ (0, 1] : lim logτdnn (x) = η , where β > 1, γ > 0 and τ > 1, η > 0. n→∞
1. Introduction Given x ∈ (0, 1], let x = [d1 (x), d2 (x), · · · , dn (x), · · · ] denote the Engel expansion of x, i.e. 1 1 1 + + ··· + + ··· , x= d1 (x) d1 (x)d2 (x) d1 (x)d2 (x) · · · dn (x) where {d j (x), j ≥ 1} is a sequence of positive integers satisfying d1 (x) ≥ 2 and d j+1 (x) ≥ d j (x) for j ≥ 1 (see [2]). On the other hand, any {d j , j ≥ 1} of integer sequence satisfying d1 ≥ 2 and d j+1 ≥ d j for j ≥ 1 is realizable, that is, there exists a unique x ∈ (0, 1] such that d j (x) = d j for any j ≥ 1. The Engel expansion is generated by the map T : (0, 1] → (0, 1], given by 1 1 1 T (x) = x − , , n ≥ 1, (n + 1), if x ∈ n+1 n+1 n and that T has no finite invariant measure equivalent to the Lebesgue measure (see [2], Chapter V). In general, rational numbers have a finite Engel expansion, while irrational numbers have an infinite Engel expansion. For example, 1.175 = [1, 6, 20]; π = [1, 1, 1, 8, 8, 17, 19, 300, 1991, 2492, · · · ]; √ 2 = [1, 3, 5, 5, 16, 18, 78, 102, 120, 144, · · · ]. In [2], J. Galambos proved that for almost all x ∈ (0, 1], 1
lim dnn (x) = e,
n→∞
2000 Mathematics Subject Classification. 11K55, 28A80 . Key words and phrases. Engel expansions; exceptional sets; Hausdorff dimension. 1
2
¨ 1,∗ , JIA LIU 2 MEIYING LU
that is, for almost all x ∈ (0, 1], log dn (x) = 1. n He conjectured that the Hausdorff dimension of the set of points that violate the above large law of numbers is one. In 2000, J. Wu [5] proved this conjecture. In another paper, Y. Y. Liu and J. Wu [3] further showed that for any α ≥ 1, the set 1 A(α) = x ∈ (0, 1] : lim dnn (x) = α lim
n→∞
n→∞
has Hausdorff dimension 1. For more studies of exceptional sets in Engel expansions, see works [4, 6]. In this note, we would like to know the size of the sets of points for which log dn (x) has super-linear growth rate. Namely, we consider the following set log dn (x) E(φ) = x ∈ (0, 1] : lim =1 , n→∞ φ(n) where φ : N → R+ is a function satisfying φ(n + 1) − φ(n) → ∞ as n → ∞. Denote by dimH the Hausdorff dimension, we prove that Theorem 1.1. Suppose φ : N → R+ is a function satisfying φ(n + 1) − φ(n) → ∞ as n → ∞, then 1 φ(n + 1) dimH E(φ) = , where b = lim sup n . 1+b n→∞ j=1 φ( j) As the applications of Theorem1.1, we also consider the cases for which log dn (x) grows with polynomial and exponential rate. Denote log dn (x) = γ , (β > 1, γ > 0); B(β, γ) = x ∈ (0, 1] : lim n→∞ nβ log dn (x) = η , (τ > 1, η > 0). C(τ, η) = x ∈ (0, 1] : lim n→∞ τn We obtain the following results, Corollary 1.2. For any β > 1 and γ > 0, dimH B(β, γ) = 1. 1 Corollary 1.3. For any τ > 1 and η > 0, dimH C(τ, η) = . τ 2. Proofs of the results In this section, we give the proofs of the results. We first collect some basic properties possessed by the Engel expansions, see Galambos [2]. Lemma 2.1 ( [2]). Suppose that {d1 , · · · , dn } ⊂ N(n ≥ 1) satisfying d1 ≥ 2, d j+1 ≥ d j for all j ≥ 1, call In (d1 , · · · , dn ) = {x ∈ [0, 1) : d1 (x) = d1 , · · · , dn (x) = dn }
HAUSDORFF DIMENSIONS OF SOME EXCEPTIONAL SETS IN ENGEL EXPANSIONS
3
an n-th order cylinder, then |In (d1 , · · · , dn )| =
1 , d1 · · · dn−1 dn (dn − 1)
where | · | denotes the diameter of a set. Consequently, we give the mass distribution principle (see [1], Proposition 4.2), which will be used to estimate the lower bound of the Hausdorff dimension of a set. Lemma 2.2 ( [1]). Let E ⊂ (0, 1] be a Borel set and μ be a measure with μ(E) > 0. Suppose that log μ(B(x, r)) lim inf ≥ s, for all x ∈ E, n→∞ log r where B(x, r) denotes the open ball with center at x and radius r. Then dimH E ≥ s. Proof of Theorem1.1. We divide the proof into two parts. Lower bound. We first give a formula for computing the Hausdorff dimension for a class of Cantor sets related to Engel expansions. Lemma 2.3. Let {sn }n≥1 be a sequence of positive integers tending to infinity with s1 ≥ 2 and for all n ≥ 1, sn+1 ≥ N sn for some positive number N ≥ 3. Set F = {x ∈ (0, 1] : sn ≤ dn (x) < N sn , ∀n ≥ 1}. Then dimH F = lim inf n→∞
log(s1 s2 · · · sn ) . log(s1 s2 · · · sn ) + log sn+1
Proof. Let s0 be the lim inf in the statement. Set
J(d1 , d2 , · · · , dn ) := cl In+1 (d1 , · · · , dn , dn+1 ), dn+1 ≥sn+1
where sk ≤ dk < N sk for all 1 ≤ k ≤ n. Here cl stands for the closure. Then it follows that ∞
J(d1 , d2 , · · · , dn ). (2.1) F= n=1 sk ≤dk
By Lemma 2.1, we have the estimation of the length of J(d1 , d2 , · · · , dn ), N n+1 s1
2 1 1 ≤ ≤ |J(d1 , · · · , dn )| = . · · · sn sn+1 d1 , · · · , dn (sn+1 − 1) s1 · · · sn sn+1
(2.2)
Since sk → ∞ as k → ∞, then log s1 + · · · + log sn = ∞. n This, together with the definition of s0 , implies that for any s > s0 , there exists a sequence {nl : l ≥ 1} such that for all l ≥ 1, lim
n→∞
(N − 1)nl < (s1 · · · snl snl +1 )(s−s0 )/2 ,
s1 · · · snl ≤ (s1 · · · snl snl +1 )(s+s0 )/2 .
¨ 1,∗ , JIA LIU 2 MEIYING LU
4
Then, by (2.1), together with (2.2), we have |J(d1 , d2 , · · · , dnl )| s H s (F) ≤ lim inf l→∞
sk ≤dk
≤ lim inf (N − 1)nl (s1 · · · snl ) l→∞
s 2 ≤ 2s. s1 · · · snl snl +1
Since s > s0 is arbitrary, we have dimH F ≤ s0 . For the lower bound, we define a measure supported on F. Let K(d1 , d2 , · · · , dn ) denote the following closed sub-interval of (0, 1], K(d1 , d2 , · · · , dn ) := cl{x ∈ (0, 1] : d1 (x) = d1 , · · · , dn (x) = dn }, where sk ≤ dk < N sk for all 1 ≤ k ≤ n, and call K(d1 , d2 , · · · , dn ) the n-th basic interval. By Lemma 2.1, we have the estimation of the length of K(d1 , d2 , · · · , dn ), N n+1 s1
1 2 1 ≤ ≤ |K(d1 , · · · , dn )| = , (2.3) 2 · · · sn−1 sn d1 · · · dn−1 dn (dn − 1) s1 · · · sn−1 s2n
and from the definition of K(d1 , d2 , · · · , dn ), it follows that ∞
F= K(d1 , d2 , · · · , dn ).
(2.4)
n=1 sk ≤dk
Let μ be a measure supported on F such that for any basic interval K(d1 , d2 , · · · , dn ) of order n, n
1 . μ(K(d1 , d2 , · · · , dn )) = (N − 1)s j j=1 By Kolmogorov’s extension theorem, μ can be extended to a probability measure supported on F. In the following, we will check the mass distribution principle with this measure. Fix s < s0 . By the definition of s0 and the fact that sk → ∞(k → ∞) and N is a constant, there exists an integer n0 such that for all n ≥ n0 , n n s
(N − 1)sk ≥ sn+1 N sk . (2.5) k=1
k=1
We take
1 . N n0 +1 (s1 · · · sn0 −1 )s2n0 For any x ∈ F, there exists an infinite sequence {d1 , d2 , · · · } with sk ≤ dk < N sk , for all k ≥ 1 such that x ∈ K(d1 , d2 , · · · , dn ) for all n ≥ 1. For any r < r0 , there exists an integer n ≥ n0 such that r0 =
N n+2 (s
1 1 ≤ r < n+1 . 2 N (s1 · · · sn−1 )s2n 1 · · · sn )sn+1
(2.6)
By (2.3) and (2.6), B(x, r) can intersect at most three n-order basic intervals or at most cn (n + 1)-order basic intervals, where cn = 2rN n+2 (s1 · · · sn )s2n+1 .
HAUSDORFF DIMENSIONS OF SOME EXCEPTIONAL SETS IN ENGEL EXPANSIONS
5
As a consequence,
μ(B(x, r)) ≤ min 3μ(K(d1 , d2 , · · · , dn )), 2rN ≤
n
k=1
n+2
(s1 · · ·
sn )s2n+1
n+1
k=1
1 (N − 1)sk .
1 1 min 3, 2rN n+2 (s1 · · · sn )s2n+1 (N − 1)sk (N − 1)sn+1
By (2.5) and the elementary inequality min{a, b} ≤ a1−s b s which holds for any a, b > 0 and 0 < s < 1, s s 1 1 1−s n+2 2 3 (s · · · s )s ≤ 4Nr s . μ(B(x, r)) ≤ 2rN 1 n n+1 sn+1 (N n s1 · · · sn ) (N − 1)sn+1 By the mass distribution principle, we have dimH F ≥ s0 . Since φ(n + 1) − φ(n) → ∞ as n → ∞, we have eφ(n+1) ≥ 3eφ(n) ,
for sufficiently large n.
Without loss of generality, we suppose eφ(n+1) ≥ 3eφ(n) for any n ≥ 1. Let E = {x ∈ (0, 1] : eφ(n) ≤ dn (x) < 3eφ(n) , ∀n ≥ 1}, then E ⊂ E(φ). Applying Lemma 2.3, we have φ(1) + · · · + φ(n) 1 ≥ . dimH E(φ) ≥ lim inf n→∞ φ(1) + · · · + φ(n) + φ(n + 1) 1+b Upper bound. Since φ(n + 1) − φ(n) → ∞, choose > 0, then there exist n0 , for any n ≥ n0 , φ(n + 1) 1 + eφ(n)(1+ ) > eφ(n)(1− ) + 1, > . (2.7) φ(n) 1− Take Ln = eφ(n)(1− ) , Mn = eφ(n)(1+ ) . The inequalities (2.7) implie that Mn > Ln + 1
and
Ln+1 > Mn ,
for any n > n0 .
Without loss of generality, we suppose L1 ≥ 2, Mn > Ln + 1 and Ln+1 > Mn for any n ≥ 1. Define F N = {x ∈ (0, 1] : Ln ≤ dn (x) ≤ Mn , ∀n ≥ N}. Fix x ∈ E(φ), if n large enough, we have φ(n)(1 − ) ≤ log dn (x) ≤ φ(n)(1 + ). That is eφ(n)(1− ) ≤ dn (x) ≤ eφ(n)(1+ ) for large enough n, therefore ∞
FN . E(φ) ⊂ N=1
¨ 1,∗ , JIA LIU 2 MEIYING LU
6
We can only estimate the upper bound of dimH F1 , Since F N can be written as a countable union of sets with the same form as F1 , then dimH F N = dimH F1 by the σ-stability of Hausdorff dimension. For any n ≥ 1, define Dn = {(σ1 , · · · , σn ) ∈ Nn : Lk ≤ σk ≤ Mk , 1 ≤ k ≤ n}. By the definition of sequences {Ln , n ≥ 1} and {Mn , n ≥ 1}, we have the sequence {σk , k ≥ 1} of integers satisfies σ1 ≥ 2 and σk+1 ≥ σk for any k ≥ 1. It follows that
F1 = J(σ1 , · · · , σn ), n≥1 (σ1 ,··· ,σn )∈Dn
where J(σ1 , · · · , σn ) = cl
In+1 (σ1 , · · · , σn , σn+1 ),
σn+1 ≥Ln+1
called an admissible cylinder of order n. Actually, all J(σ1 , · · · , σn ) with (σ1 , · · · , σn ) ∈ Dn is a cover of F1 , we could use this cover to estimate the upper bound of Hausdorff dimension of F1 . So, we need estimate the diameter of J(σ1 , · · · , σn ) and the number of all J(σ1 , · · · , σn ) in the cover. By Lemma2.1, we have 2 := Rn . |J(σ1 , · · · , σn )| ≤ L1 · · · Ln Ln+1 The number of J(σ1 , · · · , σn )’s is equal to n n
Nn := (Mk − Lk + 1) ≤ Mk . k=1
k=1
So log Nn n→∞ − log Rn log M1 + · · · + log Mn ≤ lim inf n→∞ log L1 + · · · + log Ln + log Ln+1 [φ(1) + · · · + φ(n)](1 + ) ≤ lim inf n→∞ [φ(1) + · · · + φ(n) + φ(n + 1)](1 − ) 1+ . ≤ (1 + b)(1 − )
dimH F1 ≤ lim inf
letting → 0, we get dimH E(φ) ≤
1 . 1+b
Proof of Corollary1.2. Let φ(n) = γnβ , then φ(n + 1) − φ(n) → ∞ as n → ∞. Applying Theorem1.1, we have dimH B(β, γ) = 1. Proof of Corollary1.3. Let φ(n) = ητn , then φ(n + 1) − φ(n) → ∞ as n → ∞. 1 Applying Theorem1.1, we get dimH C(τ, η) = . τ
HAUSDORFF DIMENSIONS OF SOME EXCEPTIONAL SETS IN ENGEL EXPANSIONS
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Acknowledgements. This work was supported by National Natural Science Foundation of China (No.11401066; No.11626030; No.11701001), Scientific and Technological Research Program of Chongqing Municipal Education Commission (Grant No.KJ1703046), Projects from Chongqing Municipal Science and Technology Commission (No.cstc2015jcyjA00026). References [1] K. J. Falconer, Fractal Geometry, Mathematical Foundations and Application, John Wiley and Sons, Chichester, 1990. [2] J. Galambos, Reprentations of Real Numbers by Infinite Series, Lecture Notes in Math. 502, Springer, 1976. [3] Y. Y. Liu and J. Wu, Hausdorff dimensions in Engel expansions, Acta Arith., 99(2001), 79-83. [4] Y. Y. Liu and J. Wu, Some exceptional sets in Engel expansions, Nonlinearity, 16(2)(2003), 559-566. [5] J. Wu, A problem of Galambos on Engel expansions, Acta Arith., 92(2000), 383-386. [6] J. Wu, How many points have the same Engel and Sylvester expansions, J. Number Theory, 103(2003), 16-26. 1
School of Mathematical Sciences, Chongqing Normal University, Chongqing , 401331, P. R. China E-mail address: [email protected] 2 institute of statistics and Applied Mathematics, Anhui University of Finance and Economics, Benbu, 233030, P. R. China E-mail address: [email protected]
Meiying Lü, Jia Liu
PII: DOI: Reference:
S0022-314X(17)30356-6 https://doi.org/10.1016/j.jnt.2017.09.015 YJNTH 5885
To appear in:
Journal of Number Theory
Received date: Revised date: Accepted date:
18 April 2017 18 September 2017 24 September 2017
Please cite this article in press as: M. Lü, J. Liu, Hausdorff dimensions of some exceptional sets in Engel expansions, J. Number Theory (2018), https://doi.org/10.1016/j.jnt.2017.09.015
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
HAUSDORFF DIMENSIONS OF SOME EXCEPTIONAL SETS IN ENGEL EXPANSIONS ¨ 1,∗ , JIA LIU 2 MEIYING LU Abstract. Given any real number x ∈ (0, 1], denote its Engel expansion by ∞ 1 n=1 d1 (x)···dn (x) , where {d j (x), j ≥ 1} is a sequence of positive integers satisfying d1 (x) ≥ 2 and d j+1 (x) ≥ d j (x)( j ≥ 1). Suppose φ : N → R+ is a function satisfying φ(n + 1) − φ(n) → ∞ as n → ∞. In this paper, we consider the set log dn (x) =1 , E(φ) = x ∈ (0, 1] : lim n→∞ φ(n) and we quantify the size of E(φ) in the sense of Hausdorff dimension. As appli cations, we get the Hausdorff dimensions of the sets x ∈ (0, 1] : lim logndβn (x) = γ n→∞ and x ∈ (0, 1] : lim logτdnn (x) = η , where β > 1, γ > 0 and τ > 1, η > 0. n→∞
1. Introduction Given x ∈ (0, 1], let x = [d1 (x), d2 (x), · · · , dn (x), · · · ] denote the Engel expansion of x, i.e. 1 1 1 + + ··· + + ··· , x= d1 (x) d1 (x)d2 (x) d1 (x)d2 (x) · · · dn (x) where {d j (x), j ≥ 1} is a sequence of positive integers satisfying d1 (x) ≥ 2 and d j+1 (x) ≥ d j (x) for j ≥ 1 (see [2]). On the other hand, any {d j , j ≥ 1} of integer sequence satisfying d1 ≥ 2 and d j+1 ≥ d j for j ≥ 1 is realizable, that is, there exists a unique x ∈ (0, 1] such that d j (x) = d j for any j ≥ 1. The Engel expansion is generated by the map T : (0, 1] → (0, 1], given by 1 1 1 T (x) = x − , , n ≥ 1, (n + 1), if x ∈ n+1 n+1 n and that T has no finite invariant measure equivalent to the Lebesgue measure (see [2], Chapter V). In general, rational numbers have a finite Engel expansion, while irrational numbers have an infinite Engel expansion. For example, 1.175 = [1, 6, 20]; π = [1, 1, 1, 8, 8, 17, 19, 300, 1991, 2492, · · · ]; √ 2 = [1, 3, 5, 5, 16, 18, 78, 102, 120, 144, · · · ]. In [2], J. Galambos proved that for almost all x ∈ (0, 1], 1
lim dnn (x) = e,
n→∞
2000 Mathematics Subject Classification. 11K55, 28A80 . Key words and phrases. Engel expansions; exceptional sets; Hausdorff dimension. 1
2
¨ 1,∗ , JIA LIU 2 MEIYING LU
that is, for almost all x ∈ (0, 1], log dn (x) = 1. n He conjectured that the Hausdorff dimension of the set of points that violate the above large law of numbers is one. In 2000, J. Wu [5] proved this conjecture. In another paper, Y. Y. Liu and J. Wu [3] further showed that for any α ≥ 1, the set 1 A(α) = x ∈ (0, 1] : lim dnn (x) = α lim
n→∞
n→∞
has Hausdorff dimension 1. For more studies of exceptional sets in Engel expansions, see works [4, 6]. In this note, we would like to know the size of the sets of points for which log dn (x) has super-linear growth rate. Namely, we consider the following set log dn (x) E(φ) = x ∈ (0, 1] : lim =1 , n→∞ φ(n) where φ : N → R+ is a function satisfying φ(n + 1) − φ(n) → ∞ as n → ∞. Denote by dimH the Hausdorff dimension, we prove that Theorem 1.1. Suppose φ : N → R+ is a function satisfying φ(n + 1) − φ(n) → ∞ as n → ∞, then 1 φ(n + 1) dimH E(φ) = , where b = lim sup n . 1+b n→∞ j=1 φ( j) As the applications of Theorem1.1, we also consider the cases for which log dn (x) grows with polynomial and exponential rate. Denote log dn (x) = γ , (β > 1, γ > 0); B(β, γ) = x ∈ (0, 1] : lim n→∞ nβ log dn (x) = η , (τ > 1, η > 0). C(τ, η) = x ∈ (0, 1] : lim n→∞ τn We obtain the following results, Corollary 1.2. For any β > 1 and γ > 0, dimH B(β, γ) = 1. 1 Corollary 1.3. For any τ > 1 and η > 0, dimH C(τ, η) = . τ 2. Proofs of the results In this section, we give the proofs of the results. We first collect some basic properties possessed by the Engel expansions, see Galambos [2]. Lemma 2.1 ( [2]). Suppose that {d1 , · · · , dn } ⊂ N(n ≥ 1) satisfying d1 ≥ 2, d j+1 ≥ d j for all j ≥ 1, call In (d1 , · · · , dn ) = {x ∈ [0, 1) : d1 (x) = d1 , · · · , dn (x) = dn }
HAUSDORFF DIMENSIONS OF SOME EXCEPTIONAL SETS IN ENGEL EXPANSIONS
3
an n-th order cylinder, then |In (d1 , · · · , dn )| =
1 , d1 · · · dn−1 dn (dn − 1)
where | · | denotes the diameter of a set. Consequently, we give the mass distribution principle (see [1], Proposition 4.2), which will be used to estimate the lower bound of the Hausdorff dimension of a set. Lemma 2.2 ( [1]). Let E ⊂ (0, 1] be a Borel set and μ be a measure with μ(E) > 0. Suppose that log μ(B(x, r)) lim inf ≥ s, for all x ∈ E, n→∞ log r where B(x, r) denotes the open ball with center at x and radius r. Then dimH E ≥ s. Proof of Theorem1.1. We divide the proof into two parts. Lower bound. We first give a formula for computing the Hausdorff dimension for a class of Cantor sets related to Engel expansions. Lemma 2.3. Let {sn }n≥1 be a sequence of positive integers tending to infinity with s1 ≥ 2 and for all n ≥ 1, sn+1 ≥ N sn for some positive number N ≥ 3. Set F = {x ∈ (0, 1] : sn ≤ dn (x) < N sn , ∀n ≥ 1}. Then dimH F = lim inf n→∞
log(s1 s2 · · · sn ) . log(s1 s2 · · · sn ) + log sn+1
Proof. Let s0 be the lim inf in the statement. Set
J(d1 , d2 , · · · , dn ) := cl In+1 (d1 , · · · , dn , dn+1 ), dn+1 ≥sn+1
where sk ≤ dk < N sk for all 1 ≤ k ≤ n. Here cl stands for the closure. Then it follows that ∞
J(d1 , d2 , · · · , dn ). (2.1) F= n=1 sk ≤dk
By Lemma 2.1, we have the estimation of the length of J(d1 , d2 , · · · , dn ), N n+1 s1
2 1 1 ≤ ≤ |J(d1 , · · · , dn )| = . · · · sn sn+1 d1 , · · · , dn (sn+1 − 1) s1 · · · sn sn+1
(2.2)
Since sk → ∞ as k → ∞, then log s1 + · · · + log sn = ∞. n This, together with the definition of s0 , implies that for any s > s0 , there exists a sequence {nl : l ≥ 1} such that for all l ≥ 1, lim
n→∞
(N − 1)nl < (s1 · · · snl snl +1 )(s−s0 )/2 ,
s1 · · · snl ≤ (s1 · · · snl snl +1 )(s+s0 )/2 .
¨ 1,∗ , JIA LIU 2 MEIYING LU
4
Then, by (2.1), together with (2.2), we have |J(d1 , d2 , · · · , dnl )| s H s (F) ≤ lim inf l→∞
sk ≤dk
≤ lim inf (N − 1)nl (s1 · · · snl ) l→∞
s 2 ≤ 2s. s1 · · · snl snl +1
Since s > s0 is arbitrary, we have dimH F ≤ s0 . For the lower bound, we define a measure supported on F. Let K(d1 , d2 , · · · , dn ) denote the following closed sub-interval of (0, 1], K(d1 , d2 , · · · , dn ) := cl{x ∈ (0, 1] : d1 (x) = d1 , · · · , dn (x) = dn }, where sk ≤ dk < N sk for all 1 ≤ k ≤ n, and call K(d1 , d2 , · · · , dn ) the n-th basic interval. By Lemma 2.1, we have the estimation of the length of K(d1 , d2 , · · · , dn ), N n+1 s1
1 2 1 ≤ ≤ |K(d1 , · · · , dn )| = , (2.3) 2 · · · sn−1 sn d1 · · · dn−1 dn (dn − 1) s1 · · · sn−1 s2n
and from the definition of K(d1 , d2 , · · · , dn ), it follows that ∞
F= K(d1 , d2 , · · · , dn ).
(2.4)
n=1 sk ≤dk
Let μ be a measure supported on F such that for any basic interval K(d1 , d2 , · · · , dn ) of order n, n
1 . μ(K(d1 , d2 , · · · , dn )) = (N − 1)s j j=1 By Kolmogorov’s extension theorem, μ can be extended to a probability measure supported on F. In the following, we will check the mass distribution principle with this measure. Fix s < s0 . By the definition of s0 and the fact that sk → ∞(k → ∞) and N is a constant, there exists an integer n0 such that for all n ≥ n0 , n n s
(N − 1)sk ≥ sn+1 N sk . (2.5) k=1
k=1
We take
1 . N n0 +1 (s1 · · · sn0 −1 )s2n0 For any x ∈ F, there exists an infinite sequence {d1 , d2 , · · · } with sk ≤ dk < N sk , for all k ≥ 1 such that x ∈ K(d1 , d2 , · · · , dn ) for all n ≥ 1. For any r < r0 , there exists an integer n ≥ n0 such that r0 =
N n+2 (s
1 1 ≤ r < n+1 . 2 N (s1 · · · sn−1 )s2n 1 · · · sn )sn+1
(2.6)
By (2.3) and (2.6), B(x, r) can intersect at most three n-order basic intervals or at most cn (n + 1)-order basic intervals, where cn = 2rN n+2 (s1 · · · sn )s2n+1 .
HAUSDORFF DIMENSIONS OF SOME EXCEPTIONAL SETS IN ENGEL EXPANSIONS
5
As a consequence,
μ(B(x, r)) ≤ min 3μ(K(d1 , d2 , · · · , dn )), 2rN ≤
n
k=1
n+2
(s1 · · ·
sn )s2n+1
n+1
k=1
1 (N − 1)sk .
1 1 min 3, 2rN n+2 (s1 · · · sn )s2n+1 (N − 1)sk (N − 1)sn+1
By (2.5) and the elementary inequality min{a, b} ≤ a1−s b s which holds for any a, b > 0 and 0 < s < 1, s s 1 1 1−s n+2 2 3 (s · · · s )s ≤ 4Nr s . μ(B(x, r)) ≤ 2rN 1 n n+1 sn+1 (N n s1 · · · sn ) (N − 1)sn+1 By the mass distribution principle, we have dimH F ≥ s0 . Since φ(n + 1) − φ(n) → ∞ as n → ∞, we have eφ(n+1) ≥ 3eφ(n) ,
for sufficiently large n.
Without loss of generality, we suppose eφ(n+1) ≥ 3eφ(n) for any n ≥ 1. Let E = {x ∈ (0, 1] : eφ(n) ≤ dn (x) < 3eφ(n) , ∀n ≥ 1}, then E ⊂ E(φ). Applying Lemma 2.3, we have φ(1) + · · · + φ(n) 1 ≥ . dimH E(φ) ≥ lim inf n→∞ φ(1) + · · · + φ(n) + φ(n + 1) 1+b Upper bound. Since φ(n + 1) − φ(n) → ∞, choose > 0, then there exist n0 , for any n ≥ n0 , φ(n + 1) 1 + eφ(n)(1+ ) > eφ(n)(1− ) + 1, > . (2.7) φ(n) 1− Take Ln = eφ(n)(1− ) , Mn = eφ(n)(1+ ) . The inequalities (2.7) implie that Mn > Ln + 1
and
Ln+1 > Mn ,
for any n > n0 .
Without loss of generality, we suppose L1 ≥ 2, Mn > Ln + 1 and Ln+1 > Mn for any n ≥ 1. Define F N = {x ∈ (0, 1] : Ln ≤ dn (x) ≤ Mn , ∀n ≥ N}. Fix x ∈ E(φ), if n large enough, we have φ(n)(1 − ) ≤ log dn (x) ≤ φ(n)(1 + ). That is eφ(n)(1− ) ≤ dn (x) ≤ eφ(n)(1+ ) for large enough n, therefore ∞
FN . E(φ) ⊂ N=1
¨ 1,∗ , JIA LIU 2 MEIYING LU
6
We can only estimate the upper bound of dimH F1 , Since F N can be written as a countable union of sets with the same form as F1 , then dimH F N = dimH F1 by the σ-stability of Hausdorff dimension. For any n ≥ 1, define Dn = {(σ1 , · · · , σn ) ∈ Nn : Lk ≤ σk ≤ Mk , 1 ≤ k ≤ n}. By the definition of sequences {Ln , n ≥ 1} and {Mn , n ≥ 1}, we have the sequence {σk , k ≥ 1} of integers satisfies σ1 ≥ 2 and σk+1 ≥ σk for any k ≥ 1. It follows that
F1 = J(σ1 , · · · , σn ), n≥1 (σ1 ,··· ,σn )∈Dn
where J(σ1 , · · · , σn ) = cl
In+1 (σ1 , · · · , σn , σn+1 ),
σn+1 ≥Ln+1
called an admissible cylinder of order n. Actually, all J(σ1 , · · · , σn ) with (σ1 , · · · , σn ) ∈ Dn is a cover of F1 , we could use this cover to estimate the upper bound of Hausdorff dimension of F1 . So, we need estimate the diameter of J(σ1 , · · · , σn ) and the number of all J(σ1 , · · · , σn ) in the cover. By Lemma2.1, we have 2 := Rn . |J(σ1 , · · · , σn )| ≤ L1 · · · Ln Ln+1 The number of J(σ1 , · · · , σn )’s is equal to n n
Nn := (Mk − Lk + 1) ≤ Mk . k=1
k=1
So log Nn n→∞ − log Rn log M1 + · · · + log Mn ≤ lim inf n→∞ log L1 + · · · + log Ln + log Ln+1 [φ(1) + · · · + φ(n)](1 + ) ≤ lim inf n→∞ [φ(1) + · · · + φ(n) + φ(n + 1)](1 − ) 1+ . ≤ (1 + b)(1 − )
dimH F1 ≤ lim inf
letting → 0, we get dimH E(φ) ≤
1 . 1+b
Proof of Corollary1.2. Let φ(n) = γnβ , then φ(n + 1) − φ(n) → ∞ as n → ∞. Applying Theorem1.1, we have dimH B(β, γ) = 1. Proof of Corollary1.3. Let φ(n) = ητn , then φ(n + 1) − φ(n) → ∞ as n → ∞. 1 Applying Theorem1.1, we get dimH C(τ, η) = . τ
HAUSDORFF DIMENSIONS OF SOME EXCEPTIONAL SETS IN ENGEL EXPANSIONS
7
Acknowledgements. This work was supported by National Natural Science Foundation of China (No.11401066; No.11626030; No.11701001), Scientific and Technological Research Program of Chongqing Municipal Education Commission (Grant No.KJ1703046), Projects from Chongqing Municipal Science and Technology Commission (No.cstc2015jcyjA00026). References [1] K. J. Falconer, Fractal Geometry, Mathematical Foundations and Application, John Wiley and Sons, Chichester, 1990. [2] J. Galambos, Reprentations of Real Numbers by Infinite Series, Lecture Notes in Math. 502, Springer, 1976. [3] Y. Y. Liu and J. Wu, Hausdorff dimensions in Engel expansions, Acta Arith., 99(2001), 79-83. [4] Y. Y. Liu and J. Wu, Some exceptional sets in Engel expansions, Nonlinearity, 16(2)(2003), 559-566. [5] J. Wu, A problem of Galambos on Engel expansions, Acta Arith., 92(2000), 383-386. [6] J. Wu, How many points have the same Engel and Sylvester expansions, J. Number Theory, 103(2003), 16-26. 1
School of Mathematical Sciences, Chongqing Normal University, Chongqing , 401331, P. R. China E-mail address: [email protected] 2 institute of statistics and Applied Mathematics, Anhui University of Finance and Economics, Benbu, 233030, P. R. China E-mail address: [email protected]