ACTA ARITHMETICA XCII.4 (2000)

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ACTA ARITHMETICA XCII.4 (2000)

A problem of Galambos on Engel expansions

by

Jun Wu (Wuhan)

1. Introduction. Given x in (0, 1], let x = [d

1

(x), d

2

(x), . . .] denote the Engel expansion of x, that is,

(1) x = 1

d

₁

(x) + 1

d

₁

(x)d

₂

(x) + . . . + 1

d

₁

(x)d

₂

(x) . . . d

_n

(x) + . . . ,

where {d

_j

(x), j ≥ 1} is a sequence of positive integers satisfying d

₁

(x) ≥ 2 and d

j+1

(x) ≥ d

j

(x) for j ≥ 1. (See [3].) In [3], J´anos Galambos proved that for almost all x ∈ (0, 1],

(2) lim

n→∞

d

^1/n_n

(x) = e.

He conjectured ([3], P132) that the Hausdorff dimension of the set where (2) fails is one. In this paper, we prove this conjecture:

Theorem. dim

_H

{x ∈ (0, 1] : (2) fails} = 1.

We use L

¹

to denote the one-dimensional Lebesgue measure on (0, 1] and dim

H

to denote the Hausdorff dimension.

2. Proof of Theorem. The aim of this section is to prove the main result of this paper.

By Egoroff’s Theorem, there exists a Borel set A ⊂ (0, 1] with L

¹

(A)

≥ 1/2 such that {d

^1/nn

(x), n ≥ 1} converges to e uniformly on A. In partic- ular, there exists a positive number N such that

(3) 2 ≤ d

^1/n_n

(x) ≤ 3 for all n ≥ N and x ∈ A.

Choose a positive integer M satisfying M ≥ N . For any x = [d

₁

, d

₂

, . . .] = [d

₁

, d

₂

, . . . , d

_M

, d

_{M +1}

, d

_{M +2}

, . . . , d

_{kM +1}

, d

_{kM +2}

, . . . , d

_(k+1)M

, . . .], we cons- truct a new point x ∈ (0, 1] as follows:

x = [d

₁

, d

₂

, . . .],

2000 Mathematics Subject Classification: Primary 11K55; Secondary 28A80.

Project supported by National Natural Science Foundation of China.

[383]

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384 J. Wu

where d

_{k(M +1)+l}

= d

kM +l

for all k ≥ 0 and 0 ≤ l ≤ M . That is, (4) x = [d

₁

, d

₂

, . . . , d

_M

, d

_M

, d

_{M +1}

, d

_{M +2}

, . . . ,

d

kM +1

, d

kM +2

, . . . , d

_(k+1)M

, d

_(k+1)M

, . . .].

Lemma 1. {x : x ∈ A} ⊂ {x ∈ (0, 1] : (2) fails}.

P r o o f. Note that for any k ≥ 1, d

_{k(M +1)}

(x) = d

kM

(x). We have (5) lim

k→∞

d

1/(k(M +1))

k(M +1)

(x) = lim

k→∞

(d

^{1/(kM )}_kM

(x))

kM/(k(M +1))

= e

^{M/(M +1)}

, and this proves the assertion.

For any x = [d

₁

(x), d

₂

(x), . . .] ∈ (0, 1], y = [d

₁

(y), d

₂

(y), . . .] ∈ (0, 1], define

%(x, y) = inf{j : d

j

(x) 6= d

j

(y)} (inf ∅ = ∞).

For any x, y ∈ (0, 1], x 6= y, suppose %(x, y) = k. Without loss of generality, assume d

_k

(x) < d

_k

(y). Then x > y and x ∈ (B, C], y ∈ (D, E]

with

B = 1

d

1

(x) + 1

d

1

(x)d

2

(x) + . . . + 1

d

1

(x)d

2

(x) . . . d

k−1

(x)d

k

(x)

+ 1

d

₁

(x)d

₂

(x) . . . d

_k

(x)d

_k+1

(x) ,

C = 1

d

1

(x) + 1

d

1

(x)d

2

(x) + . . . + 1

d

1

(x)d

2

(x) . . . d

k−1

(x)(d

k

(x) − 1) ,

D = 1

d

₁

(y) + 1

d

₁

(y)d

₂

(y) + . . . + 1

d

₁

(y)d

₂

(y) . . . d

_k−1

(y)d

_k

(y) ,

E = 1

d

1

(y) + 1

d

1

(y)d

2

(y) + . . . + 1

d

1

(y)d

2

(y) . . . d

k−1

(y)d

k

(y)

+ 1

d

₁

(y)d

₂

(y) . . . d

_k

(y)(d

_k+1

(y) − 1) , hence

(6) 1

d

₁

(x)d

₂

(x) . . . d

_k

(x)d

_k+1

(x) ≤ |x − y| ≤ 1

d

₁

(x)d

₂

(x) . . . d

_k−1

(x) , where d

₀

(x) ≡ 1.

Let

ε = 6 log 3

M log 2 , c = 1 3

^{4M (M +1)}

. Lemma 2. For any x, y ∈ A,

(7) |x − y| ≥ c|x − y|

^1+2ε

.

P r o o f. Without loss of generality, assume x > y. Suppose %(x, y) = k.

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Galambos on Engel expansions 385

(a) If k ≤ 2M , then by (3), (4) and (6), we have

|x−y| ≥ 1

d

₁

(x)d

₂

(x) . . . d

_k

(x)d

_k+1

(x)d

_k+2

(x) ≥

1 3

^2M

_{2M +2}

≥ c|x−y|

^1+2ε

. (b) If pM < k ≤ (p + 1)M for some p ≥ 2, then by (4) and (6), we have (8) |x − y| ≥

p−1

Y

j=0

Y

_M

l=1

1 d

_{jM +l}

(x)

1 d

_{jM +M}

(x)

_k+1

Y

j=pM +1

1 d

_j

(x) . For 1 ≤ j ≤ p − 1, by (3), we have

(9) d

_{jM +M}

(x) ≤ 3

^{jM +M}

≤ 3

^2jM

≤

Y

^M

l=1

2

^{jM +l}

_ε

≤

Y

^M

l=1

d

_{jM +l}

(x)

_ε

, thus

(10)

Y

_M

l=1

1 d

_{jM +l}

(x)

1 d

_{jM +M}

(x) ≥

Y

_M

l=1

1 d

_{jM +l}

(x)

_1+ε

, 1 ≤ j ≤ p − 1.

For j = 0, (11)

Y

_M

l=1

1 d

l

(x)

1 d

M

(x) ≥ 1 3

^{M (M +1)}

. On the other hand,

d

_k

(x)d

_k+1

(x) ≤ 3

^2k+1

≤ 3

^3k

≤ (2

^M

2

^{M +1}

. . . 2

^k−1

)

^ε

(12)

≤ (d

_M

(x) . . . d

_k−1

(x))

^ε

, hence

(13) 1

d

_k

(x)d

_k+1

(x) ≥

1 d

₁

(x)d

₂

(x) . . . d

_k−1

(x)

_ε

. Combining (10), (11) and (13), we have

|x − y|

≥ 1

3

^{M (M +1)}

p−1

Y

j=1

Y

_M

l=1

1 d

_{jM +l}

(x)

_1+ε

_k−1

Y

j=pM +1

1 d

_j

(x)

1 d

_k

(x) · 1 d

_k+1

(x)

≥ 1

3

^{M (M +1)}

_k−1

Y

i=1

1 d

_i

(x)

_1+2ε

≥ c|x − y|

^1+2ε

.

Proof of Theorem. Consider a map f : A → (0, 1] defined by f (x) = x.

Note that f : A → f (A) is bijective. Lemma 2 implies that the inverse map

of f is 1/(1 + 2ε)-H¨older. By Lemma 1 and [2], Proposition 2.3, we have

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386 J. Wu

1 = dim

H

A = dim

H

[f

⁻¹

(f (A))] ≤ (1 + 2ε) dim

H

f (A)

≤ (1 + 2ε) dim

_H

{x ∈ (0, 1] : (2) fails}.

Hence

dim

H

{x ∈ (0, 1] : (2) fails} ≥ 1 1 +

¹²_M

·

^{log 3}_{log 2}

. Since M > N is arbitrary, we have

dim

_H

{x ∈ (0, 1] : (2) fails} = 1, and this completes the proof of Theorem.

References

[1] P. E r d ˝o s, A. R´en y i and P. S z ¨ u s z, On Engel’s and Sylvester’s series, Ann. Univ.

Sci. Budapest. E¨otv¨os Sect. Math. 1 (1958), 7–32.

[2] K. J. F a l c o n e r, Fractal Geometry: Mathematical Foundations and Applications, Wiley, 1990.

[3] J. G a l a m b o s, Representations of Real Numbers by Infinite Series, Lecture Notes in Math. 502, Springer, 1976.

[4] —, The Hausdorff dimension of sets related to g-expansions, Acta Arith. 20 (1972), 385–392.

[5] —, The ergodic properties of the denominators in the Oppenheim expansion of real numbers into infinite series of rationals, Quart. J. Math. Oxford Ser. (2) 21 (1970), 177–191.

Department of Mathematics and Center of Non-linear Science Wuhan University

430072, Wuhan

People’s Republic of China E-mail: wujunyu@public.wh.hb.cn

Received on 10.6.1999 (3617)