Exceptional sets of the Oppenheim expansions over the field of formal Laurent series

Finite Fields and Their Applications 42 (2016) 253–268

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Finite Fields and Their Applications www.elsevier.com/locate/ffa

Exceptional sets of the Oppenheim expansions over the field of formal Laurent series Mei-Ying Lü a , Jia Liu b,∗ , Zhen-Liang Zhang c a

School of Mathematical Sciences, Chongqing Normal University, Chongqing, 401331, PR China b Institute of Statistics and Applied Mathematics, Anhui University of Finance and Economics, Bengbu, 233030, PR China c School of Mathematical Sciences, Henan Institute of Science and Technology, Xinxiang, 453003, PR China

a r t i c l e

i n f o

Article history: Received 15 May 2016 Received in revised form 18 August 2016 Accepted 19 August 2016 Available online 5 September 2016 Communicated by Arne Winterhof MSC: 11K55 11T06 28A80

a b s t r a c t Let Fq be a finite field with q elements, Fq ((z −1 )) denote the field of all formal Laurent series with coefficients in Fq and I be the valuation Fq ((z −1 )). For any formal ∞ideal of −n Laurent series x = ∈ I, the series a 1(x) + n=ν cn z 1 ∞ r1 (a1 (x))···rn (an (x)) 1 is the Oppenheim expansion n=1 s (a (x))···s (a (x)) a (x) 1

1

n

n+1

 E(φ) =

Keywords: Oppenheim expansions Formal Laurent series Hausdorff dimension

n

of x. Suppose φ : N → R+ is a function satisfying φ(n)/n → ∞ as n → ∞. In this paper, we quantify the size, in the sense of Hausdorff dimension, of the set n−1

x ∈ I : lim

n→∞

j=0

Δj (x)

φ(n)

 =1 ,

where Δ0 (x) = deg a1 (x) and Δn (x) = deg an+1 (x) − 2 deg an (x) − deg rn (an (x)) + deg sn (an (x)) for all n ≥ 1. As applications, we investigate the cases when φ(n) are the given polynomial or exponential functions. At the end of the article, we list some special cases (including Lüroth, Engel, Sylvester

* Corresponding author. E-mail addresses: [email protected] (M.-Y. Lü), [email protected] (J. Liu), [email protected] (Z.-L. Zhang). http://dx.doi.org/10.1016/j.ffa.2016.08.002 1071-5797/© 2016 Elsevier Inc. All rights reserved.

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expansions of Laurent series and Cantor infinite products of Laurent series) to which we apply the conclusions above. © 2016 Elsevier Inc. All rights reserved.

1. Introduction Let Fq be the finite field of q elements and Fq ((z −1 )) denote the field of all formal Laurent series with coefficients in Fq . Recall that Fq [z] denote the ring of polynomials in z with coefficients in Fq . ∞  For each x = n=ν cn z −n ∈ Fq ((z −1 )), call [x] = ν≤n≤0 cn z −n ∈ Fq [z] the integral part of x and deg x = − inf{n ∈ Z : cn = 0} the degree of x, with the convention that deg 0 = −∞. Define the absolute value on Fq ((z −1 )) as x = q deg x which is a non-Archimedean absolute value. The field Fq ((z −1 )) is locally compact and complete under the metric ρ(x, y) = x − y. ∞ Denote I = {x ∈ Fq ((z −1 )) : x < 1} = {x = n=1 cn z −n : cn ∈ Fq }, which is the valuation ideal of Fq ((z −1 )). Let P be the Haar measure on Fq ((z −1 )) normalized to 1 on I. Now we recall the Oppenheim expansions of Laurent series which is introduced by A. Knopfmacher and J. Knopfmacher [5,6]. Let {rn }n≥1 and {sn }n≥1 be two sequences of nonzero polynomials over Fq satisfying the following hypothesis deg sn − deg rn ≤ 2,

for all n ≥ 1.

(1.1)

Given x ∈ Fq ((z −1 )), put a0 (x) = [x], then define A1 (x)  = x − a0 (x). Suppose An (x) 1 (n ≥ 1) is defined. If An (x) = 0, then let an (x) = An (x) and define  An+1 (x) =

An (x) −

 sn (an (x)) 1 , an (x) rn (an (x))

(1.2)

where sn (an (x)) and rn (an (x)) denote the composition of polynomials. If An (x) = 0, this recursive process stops. We call {an (x)} the digits of x. It was shown in [5,6] that the above algorithm leads to an unique finite or infinite series which converges to x (respect to ρ). This series (see below) is called the Oppenheim expansion of Laurent series of x. Theorem 1.1 ([5,6]). Every x ∈ Fq ((z −1 )) has an unique finite or infinite convergent (respect to ρ) expansion of the form ∞

r1 (a1 (x)) · · · rn (an (x)) 1 1 + , x = a0 (x) + a1 (x) n=1 s1 (a1 (x)) · · · sn (an (x)) an+1 (x)

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255

where an (x) ∈ Fq [z], a0 (x) = [x], and deg a1 (x) ≥ 1, for any n ≥ 1, deg an+1 (x) ≥ 2 deg an (x) + 1 + deg rn (an (x)) − deg sn (an (x)). Here are some special cases which were extensively studied: • • • •

Lüroth expansion: sn (z) = z(z − 1), rn (z) = 1; Engel expansion: sn (z) = z, rn (z) = 1; Sylvester expansion: sn (z) = 1, rn (z) = 1; Cantor infinite product: sn (z) = z, rn (z) = z + 1.

Denote Δ0 (x) := deg a1 (x), Δn (x) := deg an+1 (x) − 2 deg an (x) − deg rn (an (x)) + deg sn (an (x))

(n ≥ 1),

then {Δn (x)}∞ n=0 is strictly increasing sequence (see Lemma 4 in [2]). Ai-Hua Fan and Jun Wu in [1] proved that Theorem 1.2 ([1]). Suppose that the hypothesis (1.1) is satisfied. For P -almost all x ∈ I, n−1 q 1 . Δj (x) = n→∞ n q−1 j=0

lim

Theorem 1.2 implies that almost surely, the sum of Δj (x) grows linearly. By Theorem 1.2 and noting that Δj (x) ≥ 1, Ai-Hua Fan and Jun Wu also consider the following exceptional sets ⎧ ⎨

⎫ n−1 ⎬ 1 A(α) = x ∈ I : lim Δj (x) = α n→∞ n ⎩ ⎭ j=0

(α ≥ 1).

Suppose that there exists an integer L ≤ 2 such that deg sn − deg rn = L,

for any n ≥ 1.

(1.3)

Under the hypothesis (1.3), Ai-Hua Fan and Jun Wu calculated the Hausdorff dimensions of A(α). Theorem 1.3 ([1]). Suppose that the hypothesis (1.3) is satisfied: (1) If L ≤ 1, we have dimH A(α) = 1 for all α ≥ 1.

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(2) If L = 2, we have dimH A(α) = f (q, α) with f (q, α) =

α log(αq(q − 1)) + (1 − α) log((α − 1)(q − 1)) . 2α log q

Here dimH denotes the Hausdorff dimension. In this paper, we would like to know what happens when the sum of Δj (x) grows with a general functional rate, namely, we consider the size of the following set   n−1 j=0 Δj (x) =1 E(φ) = x ∈ I : lim n→∞ φ(n) where φ : N → R+ is a function satisfying φ(n)/n → ∞ as n → ∞. We prove that Theorem 1.4. Suppose that the hypothesis (1.3) is satisfied and E(φ) = ∅. (1) If L ≤ 1, we have dimH E(φ) ≥

1 , 1+γ

with γ = lim sup n→∞

φ(n + 1) . − L)n−j φ(j)

Σnj=1 (2

(2) If L = 2, we have dimH E(φ) =

1 , 1+b

with b = lim sup n→∞

φ(n + 1) . φ(n)

As applications, we obtain the following theorem. Theorem 1.5. Suppose that the hypothesis (1.3) is satisfied and the following sets are nonempty. (1) If L ≤ 0, for any τ > 2 − L and ξ > 0, we have   n−1 1 j=0 Δj (x) ; dimH x ∈ I : lim =1 ≥ n n→∞ ξτ L+τ −1 (2) If L = 1, for any β > 1 and α > 0, we have  n−1 dimH

x ∈ I : lim

n→∞

for any τ > 1 and ξ > 0, we have  dimH

j=0

αnβ

n−1

x ∈ I : lim

n→∞

Δj (x)

j=0

Δj (x)

ξτ n

 =1

= 1;

 =1

≥

1 ; τ

M.-Y. Lü et al. / Finite Fields and Their Applications 42 (2016) 253–268

(3) If L = 2, for any β > 1 and α > 0, we have  n−1 dimH

x ∈ I : lim

n−1

x ∈ I : lim

n→∞

Δj (x)

αnβ

n→∞

for any τ > 1 and ξ > 0, we have  dimH

j=0

j=0

Δj (x)

ξτ n

 =1

=

 =1

=

257

1 ; 2

1 . τ +1

At the end of this section, we give a remark on φ. Let φ : N → R+ be a function such that φ(n)/n → ∞ as n → ∞. If E(φ) = ∅, then there exists an x0 ∈ E(φ), define n−1 ¯ ¯ = φ(n) j=0 Δj (x0 ) for all n ≥ 1. Obviously, we have φ(n)/φ(n) → 1 as n → ∞, ¯ so E(φ) = E(φ). Hence, in what follows, we can always assume that φ : N → N and φ(n + 1) − φ(n) ≥ 1 once E(φ) is non-empty. 2. Preliminary We first collect some basic properties possessed by the Oppenheim expansions of Laurent series, see [1,2]. A finite sequence of polynomials {k1 , k2 , . . . , kn } ⊂ Fq [z] is said to be admissible if it satisfies the admissibility condition deg k1 ≥ 1,

deg kj+1 ≥ 2 deg kj + 1 + deg rj (kj ) − deg sj (kj )

(1 ≤ j ≤ n − 1).

We similarly define the admissibility for an infinite sequence of polynomials. It is clear that the digital sequences {an (x)}n≥1 of the formal Laurent series are just all possible admissible sequences. We use | · | to denote the diameter of a set. Lemma 2.1 ([1,2]). Suppose that {k1 , k2 , · · · , kn } ⊂ Fq [z] (n ≥ 1) is an admissible sequence. Call J(k1 , k2 , · · · , kn ) = {x ∈ I : a1 (x) = k1 , a2 (x) = k2 , · · · , an (x) = kn } an n-th order cylinder, then the set J(k1 , k2 , · · · , kn ) is a disk with center n  r1 (k1 )···rj−1 (kj−1 ) 1 s1 (k1 )···sj−1 (kj−1 ) kj , and diameter

1 k1

+

j=2

|J(k1 , k2 , · · · , kn )| = q

n−1 j=1

(deg rj (kj )−deg sj (kj ))−2 deg kn −1

.

Remark. Since the metric ρ is non-Archimedean, each point of a disk may be considered as the center of the disk. Thus if two disks intersect, then one must contain the other.

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The lower bound of dimH E(φ) is obtained by estimating the Hausdorff dimension of a homogeneous Moran subset of E(φ). Now we recall the definition and a basic dimensional result of the homogeneous Moran set, see [3,4,7,8] for details. Let {nk }k≥1 be a sequence of positive integers and {ck }k≥1 be a sequence of positive numbers satisfying nk ≥ 2, 0 < ck < 1, n1 c1 ≤ δ and nk ck ≤ 1 (k ≥ 2), where δ is some positive number. Let D=



Dk , D0 = {∅}, Dk = {(i1 , · · · , ik ) : 1 ≤ ij ≤ nj , 1 ≤ j ≤ k} .

k≥0

If σ = (σ1 , · · · , σk ) ∈ Dk , τ = (τ1 , · · · , τm ) ∈ Dm , we define the concatenation of σ and τ as σ ∗ τ = (σ1 , · · · , σk , τ1 , · · · , τm ). Let (X, d) be a metric space, suppose that J ⊂ X is a closed subset, with diameter δ > 0. A collection F = {Jσ : σ ∈ D} of closed subsets of J is said to have a homogeneous Moran structure if it satisfies: (1) J∅ = J; (2) For any k ≥ 1 and σ ∈ Dk−1 , Jσ∗1 , Jσ∗2 , · · · , Jσ∗nk are subsets of Jσ and  int(Jσ∗i ) int(Jσ∗j ) = ∅ (i = j), where intA denotes the interior of A; (3) For any k ≥ 1 and σ ∈ Dk−1 , 1 ≤ j ≤ nk , we have |Jσ∗j | = ck . |Jσ | If F is such a collection, E :=





Jσ is called a homogeneous Moran set determined

k≥1 σ∈Dk

by F. Lemma 2.2 ([7]). For the above defined homogeneous Moran set, we have dimH E ≥ lim inf k→∞

log n1 n2 · · · nk . − log c1 c2 · · · ck+1 nk+1

3. Proof of theorem Proof of Theorem 1.4. We divide the proof into two parts. Part I: L ≤ 1: We define a sequence of integers {qn , n ≥ 1} as follows: q1 = φ(1),

qn+1 = (2 − L)qn + φ(n + 1) − φ(n)

(∀n ≥ 1).

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259

Then let Dn = {σ : σ = (b1 , b2 , · · · , bn ) ∈ Fq [z]n , deg bk = qk , 1 ≤ k ≤ n}. We first show that for any (b1 , b2 , · · · , bn ) ∈ Dn , the cylinder J(b1 , b2 , · · · , bn ) is a nonempty disk. By the condition (1.3), for any a, b ∈ Fq [z] with positive degree, we have deg b − 2 deg a + deg sn (a) − deg rn (a) = deg b − (2 − L) deg a.

(3.1)

By (3.1), we can check that {b1 , b2 , · · · , bn } is admissible. In fact, deg b1 = φ(1) ≥ 1, deg bj+1 = qj+1 = (2 − L)qj + φ(j + 1) − φ(j) ≥ (2 − L)qj + 1 = 2 deg bj + deg rj (bj ) − deg sj (bj ) + 1 Thus, by Lemma 2.1, J(b1 , b2 , · · · , bn ) is the disk with center

(1 ≤ j ≤ n). 1 b1

+

n  j=2

r1 (b1 )···rj−1 (bj−1 ) 1 s1 (b1 )···sj−1 (bj−1 ) bj ,

and diameter n−1 

q

(deg rj (bj )−deg sj (bj ))−2 deg bn −1

j=1

=q

−L

n−1 

qj −2qn −1

j=1

.

Put E0 = I, En =



J(b1 , b2 , · · · , bn ),

∀ n ≥ 1.

(b1 ,b2 ,··· ,bn )∈Dn

Define E=

+∞ 

En .

n=1

From the definitions of Dn and qn , it is easy to check that En = {x ∈ I : Δ0 (x) = φ(1), · · · , Δn−1 (x) = φ(n) − φ(n − 1)}, so that   E = x ∈ I : Δ0 (x) = φ(1), Δn (x) = φ(n + 1) − φ(n) for all n ≥ 1 . Thus E ⊂ E(φ).

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Take nk = (q − 1)q qk and c1 = q −2q1 −1 , ck = q (2−L)qk−1 −2qk for all k ≥ 2. From the above structure, it follows that each component J(b1 , b2 , · · · , bk−1 ) in Ek−1 contains nk many elements J(b1 , b2 , · · · , bk ) in Ek with the same ratio ck . Thus E is a standard homogeneous Moran set. By the definition of qn , we have n

qj = (2 − L)

n−1

j=1

qj + φ(n).

j=1

This implies  n−1

qn = (1 − L)

 qj

+ φ(n),

(3.2)

j=1

and n

qj =

j=1

n

(2 − L)n−j φ(j).

(3.3)

j=1

Thus, by Lemma 2.2, notice that φ(n)/n → ∞ as n → ∞, and together with (3.2) and (3.3), we have dimH E ≥ lim inf k→∞

≥ lim inf k→∞

≥

k log(q − 1) + (q1 + · · · + qk ) log q − log(q − 1) + L(q1 + · · · + qk ) log q + qk+1 log q 1 q L + q + ·k+1 · · + qk 1 1

1 + lim sup k k→∞

≥

φ(k+1)

j=1 (2−L)

k−j φ(j)

1 . 1+γ

Since E ⊂ E(φ), we have dimH E(φ) ≥

1 . 1+γ

2

Part II: L = 2: In this case, we have Δn (x) = deg an+1 (x),

Δ0 (x) = deg a1 (x),

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261

and every sequence {b1 , b2 , · · · , bn } ⊂ Fq [z] such that deg bj ≥ 1 (1 ≤ j ≤ n) is admissible. Lower bound We first give the lower bound of the Hausdorff dimension. We define a sequence of integer {qn , n ≥ 1} as follows: q1 = φ(1),

qn+1 = φ(n + 1) − φ(n)

(∀n ≥ 1).

Let Dn = {σ : σ = (b1 , b2 , · · · , bn ) ∈ Fq [z]n , deg bk = qk , 1 ≤ k ≤ n}. We know that for any (b1 , b2 , · · · , bn ) ∈ Dn , the cylinder J(b1 , b2 , · · · , bn ) is a n  r1 (b1 )···rj−1 (bj−1 ) 1 nonempty disk with center b11 + s1 (b1 )···sj−1 (bj−1 ) bj , and diameter j=2

n−1 

q

(deg rj (bj )−deg sj (bj ))−2 deg bn −1

j=1

=q

−2

n  j=1

qj −1

.

Put E0 = I, En =



J(b1 , b2 , · · · , bn ),

∀ n ≥ 1.

(b1 ,b2 ,··· ,bn )∈Dn

Define E=

+∞ 

En .

n=1

From the definitions of Dn and qn , it is easy to check that En = {x ∈ I : Δ0 (x) = φ(1), · · · , Δn−1 (x) = φ(n) − φ(n − 1)}, so that   E = x ∈ I : Δ0 (x) = φ(1), Δn (x) = φ(n + 1) − φ(n) for all n ≥ 1 . Thus E ⊂ E(φ). Take nk = (q − 1)q qk and c1 = q −2q1 −1 , ck = q −2qk for all k ≥ 2. From the above structure, it follows that each component J(b1 , b2 , · · · , bk−1 ) in Ek−1 contains nk many elements J(b1 , b2 , · · · , bk ) in Ek with the same ratio ck . Thus E is a standard homogeneous Moran set. By Lemma 2.2, and noticing that φ(n)/n → ∞ as n → ∞, we have

M.-Y. Lü et al. / Finite Fields and Their Applications 42 (2016) 253–268

262

k log(q − 1) + φ(k) log q − log(q − 1) + (φ(k + 1) + φ(k)) log q 1

dimH E ≥ lim inf k→∞

≥ lim inf k→∞

≥

1+

φ(k+1) φ(k)

1 . 1+b

Since E ⊂ E(φ), we have 1 . 1+b

dimH E(φ) ≥

Upper bound Now we give the upper bound of the Hausdorff dimension. Since φ(n) is monotonic increasing, we have b ≥ 1, the following proof is distinguished into three cases according to 1 < b < ∞, b = 1 and b = ∞. Case I : 1 < b < ∞. For any {b1 , b2 , · · · , bn } ⊂ Fq [z] such that deg bj ≥ 1 (1 ≤ j ≤ n), J(b1 , b2 , · · · , bn ) is a disk with diameter |J(b1 , b2 , · · · , bn )| = q

−2

n  j=1

deg bj −1

.

For each t > 1, we define a probability measure μt on I by letting μt (J(b1 , b2 , · · · , bn )) = q

(−t

n 

deg bj −nP (t))

j=1

where P (t) = logq (q(q − 1)) − logq (q t − q). By using the equalities

q −t deg b =

∞ n=1 deg b=n

b:deg b≥1

q −tn =

∞

q −tn (q − 1)q n =

n=1

q(q − 1) = q p(t) , qt − q

it is easy to check that

μt (J(b1 , b2 , · · · , bn+1 )) = μt (J(b1 , b2 , · · · , bn ));

bn+1 :deg bn+1 ≥1



μt (J(b1 , b2 , · · · , bn )) = 1,

b1 ,b2 ,··· ,bn

where the sum is taken over all bj such that deg bj ≥ 1. So the measure μt is well defined. 1+ Fix t > 1,  > 0 such that b > (1−) 2 . By the given conditions φ(n)/n → ∞ as n → ∞, we have

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263

• There exists N (t, ε) ∈ N such that for all n ≥ N (t, ε), nP (t) ≤ ε(1 − ε)φ(n).

(3.4)

• We can choose a subsequence {nk }∞ k=1 of N with nk ≥ N (t, ε) for all k ≥ 1 and φ(nk + 1) ≥ φ(nk )b(1 − ε).

(3.5)

Now we give a cover of the set E(φ). For any n ≥ 1, let  Cn () =

(b1 , b2 , · · · , bn ) ∈ Fnq [z], (1 − ) ≤

 n 1 deg bj ≤ (1 + ) . φ(n) j=1

For any (b1 , b2 , · · · , bn ) ∈ Cn (), let Dn+1 (, (b1 , b2 , · · · , bn )) =

  bn+1 ∈ Fq [z], (b1 , b2 , · · · , bn+1 ) ∈ Cn+1 () ,

and 

I(b1 , b2 , · · · , bn ) =

J(b1 , b2 , · · · , bn+1 ).

bn+1 ∈Dn+1 (,(b1 ,b2 ,··· ,bn ))

Then E(φ) ⊂

∞  ∞ 



I(b1 , b2 , · · · , bn ).

N =1 n=N (b1 ,b2 ,··· ,bn )∈Cn ()

For each N ≥ 1, (b1 , b2 , · · · , bnk ) ∈ Cnk () with nk ≥ N , we will estimate the length of I(b1 , b2 , · · · , bnk ). For any bnk +1 ∈ Dnk +1 (, (b1 , b2 , · · · , bnk )), by the definition of Dnk +1 (, (b1 , b2 , · · · , bnk )) and Cnk (), together with (3.5), we have n k +1 j=1

nk b(1 − )2 deg bj ≥ φ(nk + 1)(1 − ) ≥ φ(nk )b(1 − ) ≥ deg bj . 1 +  j=1 2

Thus  deg bnk +1 ≥

Writing β =

b(1−)2 1+

 nk b(1 − )2 −1 deg bj . 1+ j=1

− 1, by Lemma 2.1, we have

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|I(b1 , b2 , · · · , bnk )| ≤

bnk +1 :deg bnk +1 ≥β ∞

≤ k= β

≤q q 2

nk

j=1

nk

|J(b1 , b2 , · · · , bnk +1 )|

j=1

deg bj

(q − 1)q

−k

q

−2



nk

deg bj −1

j=1

deg bj

−(2+β)



nk

deg bj

j=1

|J(b1 , b2 , · · · , bnk )|

2+β 2

.

Subsequently, for each (b1 , · · · , bnk ) ∈ Cnk (), by the definition of Cn () and (3.4), we have |J(b1 , b2 , · · · , bnk )|

t+ 2

(q ≤q

−2

−t



nk

deg bj

j=1



nk

)

t+ 2

deg bj −nk P (t)

j=1

After these preliminaries, we will estimate the E(φ). t+

H 2+β (E(φ)) ≤ lim inf k→∞

lim inf k→∞

lim inf k→∞

= μt (J(b1 , b2 , · · · , bnk )).

t+ 2+β -dimensional



Hausdorff measure of

t+

|I(b1 , b2 , · · · , bnk )| 2+β

(b1 ,b2 ,··· ,bnk )∈Cnk ()



|J(b1 , b2 , · · · , bnk )|

t+ 2

(b1 ,b2 ,··· ,bnk )∈Cnk ()



μt (J(b1 , b2 , · · · , bnk ))

(b1 ,b2 ,··· ,bnk )∈Cnk ()

≤ 1. So, we have dimH E(φ) ≤

t+ 2+β .

Letting  → 0 and t → 1, we obtain dimH E(φ) ≤

1 . 1+b

Case II : b = 1. In this case, it can be proved by just applying the natural covering system E(φ) ⊂

∞  ∞ 



J(b1 , b2 , · · · , bn ).

N =1 n=N (b1 ,b2 ,··· ,bn )∈Cn ()

Then, take the same steps, by the definition of Cnk () and (3.4), we have

M.-Y. Lü et al. / Finite Fields and Their Applications 42 (2016) 253–268

H

t+ 2



(E(φ)) ≤ lim inf k→∞



 nk

j=1

deg bj

q −t

nk

j=1

deg bj −nk p(t)

(b1 ,b2 ,··· ,bnk )∈Cnk ()



lim inf k→∞

q −(t+)

(b1 ,b2 ,··· ,bnk )∈Cnk ()

lim inf k→∞

t+ 2

(b1 ,b2 ,··· ,bnk )∈Cnk ()

lim inf k→∞

|J(b1 , b2 , · · · , bnk )|

265

μt (J(b1 , b2 , · · · , bnk ))

(b1 ,b2 ,··· ,bnk )∈Cnk ()

≤ 1. It follows that dimH E(φ) ≤

t+ 2 .

Letting  → 0 and t → 1, we have dimH E(φ) ≤

1 . 2

Case III : b = ∞. In this case, we replace b in case I by arbitrary large number m, then we have 1 dimH E(φ) ≤ 1+m . Letting m → ∞, we get dimH E(φ) = 0. 2 Proof of Theorem 1.5. Let φ(n) = αnβ and ξτ n respectively. By Theorem 1.4, we get the results of Theorem 1.5. 2 We now list some special cases to which we apply Theorem 1.4 and Theorem 1.5. The case of Engel expansions has been studied by Meiying Lü [7]. However, the following results on Lüroh expansions, Sylvester expansions and Cantor infinite products are original. Example 1. Let sn (z) = z(z − 1), rn (z) = 1 for all n ≥ 1. Then the algorithm (1.2) leads to a Lüroth expansion of Laurent series of x ∈ I, ∞

1 1 + . x= a1 (x) n=2 a1 (x)(a1 (x) − 1) · · · an−1 (x)(an−1 (x) − 1)an (x) In this case, L = 2 and Δn (x) = deg an+1 (x) for all n ≥ 0. By Theorem 1.4 and Theorem 1.5, we have Corollary 3.1. For Lüroth expansion of Laurent series, set  E(φ) =

n x ∈ I : lim

n→∞

j=1

deg aj (x)

φ(n)

 =1

266

M.-Y. Lü et al. / Finite Fields and Their Applications 42 (2016) 253–268

where φ : N → R+ is a function satisfying φ(n)/n → ∞ as n → ∞. If E(φ) = ∅, we have 1 , 1+b

dimH E(φ) =

φ(n + 1) . φ(n)

with b = lim sup n→∞

Corollary 3.2. Suppose the following sets are nonempty, then: • For any β > 1 and α > 0, we have  dimH

n j=1

x ∈ I : lim

deg aj (x)

αnβ

n→∞

 =1

=

1 ; 2

• For any τ > 1 and ξ > 0, we have  dimH

n x ∈ I : lim

n→∞

j=1

deg aj (x) ξτ n

 =1

=

1 . τ +1

2

Example 2. Let sn (z) = z, rn (z) = 1 for all n ≥ 1. Then the algorithm (1.2) leads to a Engel expansion of Laurent series of x ∈ I, x=

∞

1 . a (x)a (x) · · · an (x) 2 n=1 1

In this case, L = 1 and Δ0 (x) = deg a1 (x), Δn (x) = deg an+1 (x) − deg an (x) for all n ≥ 1. By Theorem 1.4 and Theorem 1.5, we have Corollary 3.3 ([7]). For Engel expansion of Laurent series, set  E(φ) =

 deg an (x) =1 n→∞ φ(n)

x ∈ I : lim

where φ : N → R+ is a function satisfying φ(n)/n → ∞ as n → ∞. If E(φ) = ∅, we have dimH E(φ) ≥

1 , 1+γ

with γ = lim sup n→∞

φ(n + 1) . Σnj=1 φ(j)

Corollary 3.4 ([7]). Suppose the following sets are nonempty, then: • For any β > 1 and α > 0, we have   deg an (x) dimH x ∈ I : lim = 1 = 1; n→∞ αnβ

M.-Y. Lü et al. / Finite Fields and Their Applications 42 (2016) 253–268

267

• For any τ > 1 and ξ > 0, we have   1 deg an (x) dimH x ∈ I : lim =1 ≥ . n n→∞ ξτ τ

2

Example 3. Let sn (z) = 1, rn (z) = 1 for all n ≥ 1. Then the algorithm (1.2) leads to a Sylvester expansion of Laurent series of x ∈ I, x=

∞

1 . a (x) n=1 n

In this case, L = 0 and Δ0 (x) = deg a1 (x), Δn (x) = deg an+1 (x) − 2 deg an (x) for all n ≥ 1. By Theorem 1.4 and Theorem 1.5, we have Corollary 3.5. For Sylvester expansion of Laurent series, set  x ∈ I : lim

E(φ) =

deg an (x) −

n−1 j=1

deg aj (x)

φ(n)

n→∞

 =1

where φ : N → R+ is a function satisfying φ(n)/n → ∞ as n → ∞. If E(φ) = ∅, we have dimH E(φ) ≥

1 , 1+γ

with γ = lim sup n→∞

φ(n + 1) . Σnj=1 2n−j φ(j)

Corollary 3.6. Suppose the following set is nonempty, for any τ > 2 and ξ > 0, we have  dimH

x ∈ I : lim

deg an (x) −

n→∞

n−1 j=1 n ξτ

deg aj (x)

 =1

≥

1 . τ −1

2

Example 4. Let sn (z) = z, rn (z) = z + 1 for all n ≥ 1. Then the algorithm (1.2) leads to a Cantor infinite product of Laurent series of x ∈ I, 1+x=

∞   n=1

 1 1+ . an (x)

In this case, L = 0 and Δ0 (x) = deg a1 (x), Δn (x) = deg an+1 (x) − 2 deg an (x) for all n ≥ 1. By Theorem 1.4 and Theorem 1.5, we have Corollary 3.7. For the Cantor infinite products of Laurent series, set  E(φ) =

x ∈ I : lim

n→∞

deg an (x) −

n−1 j=1

φ(n)

deg aj (x)

 =1

M.-Y. Lü et al. / Finite Fields and Their Applications 42 (2016) 253–268

268

where φ : N → R+ is a function satisfying φ(n)/n → ∞ as n → ∞. If E(φ) = ∅, we have dimH E(φ) ≥

1 , 1+γ

with γ = lim sup n→∞

φ(n + 1) . Σnj=1 2n−j φ(j)

Corollary 3.8. Suppose the following set is nonempty, for any τ > 2 and ξ > 0, we have  dimH

x ∈ I : lim

n→∞

deg an (x) −

n−1 j=1 ξτ n

deg aj (x)

 =1

≥

1 . τ −1

2

Acknowledgments This work was supported by National Natural Science Foundation of China (No. 11401066, No. 11401156, No. 11501168), Scientific and Technological Research Program of Chongqing Municipal Education Commission (Grant No. KJ1400535), Natural Science Foundation Projects from Chongqing Municipal Science and Technology Commission (No. cstc2015jcyjA00026). References [1] A.H. Fan, J. Wu, Metric properties and Exceptional sets of the Oppenheim expansions over the field of Laurent series, Constr. Approx. 20 (2004) 465–495. [2] A.H. Fan, J. Wu, Approximation orders of formal Laurent series by Oppenheim rational functions, J. Approx. Theory 121 (2003) 269–386. [3] D.J. Feng, Z.Y. Wen, J. Wu, Some dimensional results for homogeneous Moran sets, Sci. China Ser. A 40 (1997) 475–482. [4] X.H. Hu, B.W. Wang, J. Wu, Y.L. Yu, Cantor sets determined by partial quotients of continued fractions of Laurent series, Finite Fields Appl. 14 (2008) 417–437. [5] A. Knopfmacher, J. Knopfmacher, Inverse polynomial expansions of Laurent series, Constr. Approx. 4 (4) (1988) 379–389. [6] A. Knopfmacher, J. Knopfmacher, Inverse polynomial expansions of Laurent series, II, J. Comput. Appl. Math. 28 (1989) 249–257. [7] M.Y. Lü, A note on Engel series expansions of Laurent series and Hausdorff dimensions, J. Number Theory 142 (2014) 44–50. [8] M.Y. Lü, B.W. Wang, J. Xu, On sums of degrees of the partial quotients in continued fraction expansions of Laurent series, J. Math. Anal. Appl. 380 (2011) 807–813.