(Sometimes we write an(α) and pn(α)/qn(α) to stress the dependence on α.) It is a classic result of P

LXXXI.4 (1997)

On the distribution of

the sequence (nα) with transcendental α

by

Christoph Baxa (Wien)

1. Introduction. Let α ∈ R be irrational with regular continued frac- tion expansion α = [a0, a1, a2, . . .] (i.e. a0 ∈ Z and ai ∈ N for all i ≥ 1) and convergents p_n/q_n = [a₀, a₁, . . . , a_n]. (Sometimes we write a_n(α) and p_n(α)/q_n(α) to stress the dependence on α.) It is a classic result of P. Bohl [5], W. Sierpi´nski [15], [16] and H. Weyl [17], [18] that the sequence (nα)n≥1

is uniformly distributed modulo 1. This property is studied from a quanti- tative viewpoint by means of the speed of convergence in the limit relation limN →∞D_N^∗(α) = 0 where the quantity

D^∗_N(α) = sup

0≤x≤1

  1

N XN n=1

c_[0,x)({nα}) − x  

is called discrepancy. According to a theorem of W. M. Schmidt [11] the convergence is best possible if D_N^∗(α) = O((log N )/N ). It was first observed by H. Behnke [4] that this estimate is satisfied if and only if α is of bounded density, i.e. P_m

i=1a_i = O(m) as m → ∞. For α of bounded density the map α 7→ ν^∗(α) = lim sup_{N →∞}N D_N^∗(α)/ log N is used to obtain more detailed information. It was proved by Y. Dupain and V. T. S´os [6] that inf_α∈Bν^∗(α) = ν^∗([2]) where B denotes the set of numbers of bounded density and [2] = [2, 2, 2, . . .] = 1 +√

2 is used as a convenient shorthand notation. J. Schoißengeier [14] expressed ν^∗(α) in terms of the continued fraction expansion of α after he had obtained partial results in [13]. Em- ploying these results C. Baxa [3] showed the following:

(1) Let B^q := {α ∈ B | α is a quadratic irrationality}. Then we have ν^∗(B) = ν^∗(B^q) = [ν^∗([2]), ∞).

(2) Let b ≥ 4 be an even integer, B_b:= {α = [a₀, a₁, a₂, . . .] ∈ B | a_i≥ b for all i ≥ 1} and B_b^q := {α ∈ Bb | α is a quadratic irrationality}. Then ν^∗(B_b) = ν^∗(B^q_b) = [ν^∗([b]), ∞).

1991 Mathematics Subject Classification: 11K31, 11K38, 11J81.

[357]

It is the purpose of the present paper to strengthen these results and to prove:

Theorem 1. Let B^t := {α ∈ B | α is transcendental} and B^u := {α ∈ B | α is a U₂-number}. Then

ν^∗(B^t) = ν^∗(B^u) = [ν^∗([2]), ∞).

Theorem 2. Let B_b^t := {α ∈ B_b | α is transcendental} and B_b^u :=

{α ∈ Bb | α is a U2-number} (where again b ≥ 4 is assumed to be an even integer ). Then

ν^∗(B_b^t) = ν^∗(B_b^u) = [ν^∗([b]), ∞).

R e m a r k s. (1) For a more detailed and leisurely exposition of the prob- lem and its history the reader is referred to [3].

(2) In contrast to Theorem 1 we see that ν^∗(B^q) $ [ν^∗([2]), ∞) since ν^∗(α) is transcendental if α is a quadratic irrationality. This follows from Theorem 1 in §4 of [14] as the logarithm of an algebraic number 6= 1 is always transcendental.

2. Criteria for transcendence. Our criteria are a variant of a method used by E. Maillet [7, Chapter 7] and A. Baker [1], [2] (see also [8, §36]). We will follow rather closely parts of [1] and [2] with two major differences:

(1) We will use a theorem by W. M. Schmidt which became available only a few years later [9] and was generalized in [10] (compare also with [12]).

(2) We do not aim at criteria of great generality but at specific ones which are well suited for our purpose. This explains the special shape of Corollary 6 below.

Definition. If β is algebraic then H(β) denotes the classical absolute height. This means, if p(X) =P_m

i=0aiXⁱ∈ Z[X] \ {0} with gcd(a0, . . . , am)

= 1 and p(β) = 0 (and deg p minimal with this property) then H(β) = max

0≤i≤m|ai|.

Theorem 3 (W. M. Schmidt). Let α ∈ R be algebraic but neither rational nor a quadratic irrationality and δ > 0. Then there exist only finitely many β ∈ R which are rational or quadratic irrationalities such that |α − β| <

H(β)^−3−δ.

Corollary 4. Let α ∈ R have “quasiperiodic” but not periodic continued fraction expansion

α = [0, a₁, . . . , a_ν1−1, a_ν1, . . . , a_ν1+k1−1^λ1, a_ν2, . . . , a_ν2+k2−1^λ2, . . .]

(i.e. ν_n= ν₁+P_n−1

i=1 λ_ik_i). If α is algebraic then lim sup_i→∞q_νi+1−1q^−3−δ_ν

i+k_i−1

< ∞. (Here a_ν, . . . , a_ν+k^λ indicates that the partial quotients a_ν, . . . , a_ν+k

should be repeated λ times. For example [0, 1, 2, 3², 5³, 7, . . .] = [0, 1, 2, 3, 1, 2, 3, 5, 5, 5, 7, . . .].)

P r o o f. For i ≥ 1 we define the quadratic irrationality β_i:= [0, a₁, . . . , a_ν1−1, a_ν1, . . . , a_ν1+k1−1^λ1, . . .

. . . , a_νi−1, . . . , a_{νi−1+ki−1−1}^λi−1, a_νi, . . . , a_νi+ki−1].

For k ≤ νi+1 − 1 we have ak(α) = ak(βi) and we may write pk/qk for p_k(α)/q_k(α) = p_k(β_i)/q_k(β_i). Now

Liβ_i²+ Miβi+ Ni= 0 with

L_i= q_νi−2q_νi+ki−1− q_νi−1q_νi+ki−2,

Mi= qν_i−1pν_i+k_i−2+ pν_i−1qν_i+k_i−2− pν_i−2qν_i+k_i−1− qν_i−2pν_i+k_i−1, N_i= p_νi−2p_νi+ki−1− p_νi−1p_νi+ki−2,

and therefore

H(βi) ≤ max{|Li|, |Mi|, |Ni|} < 2q_ν²_i+ki−1. Theorem 3 implies

q⁻²_νi+1−1> |α − β_i| > C(α, δ)H(β_i)^−3−δ> C(α, δ)2^−3−δq_ν^−6−2δ

i+k_i−1

for a certain C(α, δ) > 0. The corollary follows immediately.

Lemma 5. Keeping all notations of Corollary 4 we have 0 < |L_iα²+ M_iα + N_i| < 8q_ν⁴_i+ki+1q⁻²_νi+1−1.

P r o o f. Let β_i denote the conjugate of β_i. If |β_i| ≥ 1 it follows from Liβ²_i + Miβi+ Ni= 0 that

|β_i|²≤ |L_iβ²_i| = |M_iβ_i+ N_i| < 2q_ν²_i+ki−1(|β_i| + 1)

≤ 4q_ν²_i+ki−1|βi|

and therefore |β_i| < 4q_ν²_i+ki−1, which remains true even if |β_i| < 1. This implies |α − βi| ≤ 1 + |βi| < 1 + 4q²_νi+ki−1< 8q²_νi+ki−1 and thus

|Liα²+ Miα + Ni| = |Li| · |α − βi| · |α − βi|

< q²_νi+ki−1· q_ν⁻²_i+1−1· 8q_ν²_i+ki−1= 8q⁴_νi+ki−1q⁻²_νi+1−1. Corollary 6. (1) Let b > a > 1 be integers and α = [0, a^λ1, b^λ2, a^λ3, b^λ4, . . .]. If

lim sup

n→∞



λ_n+1− 13log b

log a(λ₁+ . . . + λ_n)



= ∞ then α is transcendental.

(2) If even lim sup_n→∞λ_n+1/(λ₁+. . .+λ_n) = ∞ then α is a U₂-number.

P r o o f. If i > 1 then

qν_i+k_i−1= qν_i ≤ (b + 1)^{1+λ1+...+λi−1}

≤ (b²)^{2(λ1+...+λi−1)}= a^{4log blog a(λ1+...+λi−1)} and therefore

q_νi+1−1q_ν^−13/4

i+k_i−1≥ a^{λi−13log alog b(λ1+...+λi−1)} and (1) follows immediately from Corollary 4.

We have

H(L_iX²+ M_iX + N_i) = max{|L_i|, |M_i|, |N_i|} < 2q_ν²_i+ki−1≤ 2b^{4(νi+ki−1)} where H denotes the height of a polynomial just for once. Now estimating q_νi+ki−1≤ b^{2(νi+ki−1)} and q_νi+1−1≥ a^νi+1−1we deduce from Lemma 5 that

0 < |L_iα²+ M_iα + N_i|

< b^−(2(νi+1−1) log a−8(ν_i+k_i−1) log b−3 log 2)/ log b= (2b^{4(νi+ki−1)})^−Ψi with

Ψ_i= 2(ν_i+1− 1) log a − 8(ν_i+ k_i− 1) log b − 3 log 2 4(ν_i+ k_i− 1) log b + log 2 .

Obviously lim sup_i→∞Ψ_i= ∞ is equivalent to lim sup_i→∞ν_i+1/ν_i= ∞ and therefore to lim sup_i→∞λ_i/(λ₁+ . . . + λ_i−1) = ∞.

3. Values of ν^∗(α) for transcendental α

Lemma 7. Let a < b be even positive integers and ν^∗([a]) < µ < ν^∗([b]).

Then there exists a transcendental α = [0, a₁, a₂, . . .] (and even a U₂-number α) such that ai∈ {a, b} for all i ≥ 1 and ν^∗(α) = µ.

P r o o f. The function

fab(x) = 1

8· a + xb

log([a]) + x log([b])

increases for positive x, f_ab(0) = ν^∗([a]) and lim_x→∞f_ab(x) = ν^∗([b]).

Therefore there is a unique Q ∈ (0, ∞) such that µ = fab(Q). Let (σn)n≥1

be a strictly increasing sequence of integers such that σ₁Q ≥ 1 and

(1) lim sup

n→∞



σ_n+1− 13(Q + 1)log b

log a(σ₁+ . . . + σ_n)



= ∞ or even

(2) lim sup

n→∞

σ_n+1

σ₁+ . . . + σ_n = ∞

are satisfied. Let λ_2n−1 = 2σ_nand λ_2n= 2[σ_nQ] for n ≥ 1. Furthermore, let α = [0, a^λ1, b^λ2, a^λ3, b^λ4, . . .]. Using Corollary 6 it is easy to check that α is transcendental if (1) and a U₂-number if (2) is satisfied. Employing a special

case of Theorem 1 in §3 of [14] which was already stated as Theorem 1 in

§4 of [13] we see that ν^∗(α) = 1

4lim sup

m→∞

1 log qm

max

 X

1≤i≤m,2|i

a_i, X

1≤i≤m,2-i

a_i



= 1

8lim sup

m→∞

1 log q_m

Xm i=1

ai

where we used the fact that lim_m→∞log q_m+1/ log q_m = 1 for numbers of bounded density and that

max

 X

1≤i≤m,2|i

a_i, X

1≤i≤m,2 - i

a_i



= 1 2

Xm i=1

a_i+ ∆ with |∆| ≤ b/2.

If λ1+ . . . + λ2k−1< m ≤ λ1+ . . . + λ2k+1 then

log q_m= (λ₁+ λ₃+ . . . + λ_2k−1+ r_2k+1) log([a])

+ (λ₂+ λ₄+ . . . + λ_2k−2+ r_2k) log([b]) + O(k) with an implicit constant that depends on a and b only. Here

1 ≤ r2k = m − (λ1+ . . . + λ2k−1) ≤ λ2k, r2k+1 = 0

if m ≤ λ₁+ . . . + λ_2k, r2k = λ2k, 1 ≤ r2k+1 = m − (λ1+ . . . + λ2k) ≤ λ2k+1

if m > λ₁+ . . . + λ_2k. (If the reader considers this step to be too sketchy he or she may want to con- sult the proof of Theorem 4.3 in [3].) Therefore ν^∗(α) = ¹₈lim sup_m→∞h(m) where

h(m)

= ^{(λ1+ λ3}+ . . . + λ_2k−1+ r_2k+1)a + (λ₂+ λ₄+ . . . + λ_2k−2+ r_2k)b (λ₁+ λ₃+ . . . + λ_2k−1+ r_2k+1) log([¯a]) + (λ₂+ λ₄+ . . . + λ_2k−2+ r_2k) log([¯b]). Obviously max{h(m) | λ1+ . . . + λ2k−1< m ≤ λ1+ . . . + λ2k+1} = h(λ1+ . . . + λ_2k) and thus

ν^∗(α) = ¹₈ lim

k→∞sup

m≥k

h(m) = ¹₈ lim

k→∞ sup

m>λ1+...+λ2k−1

h(m)

= ¹₈ lim

k→∞sup

m≥k

h(λ₁+ . . . + λ_2m) = ¹₈lim sup

k→∞

h(λ₁+ . . . + λ_2k) = µ

since lim_k→∞(λ₂+ λ₄+ . . . + λ_2k)/(λ₁+ λ₃+ . . . + λ_2k−1) = Q.

Lemma 8. Let a be an even positive integer. Then there exists a transcen- dental α = [0, a₁, a₂, . . .] (and even a U₂-number α) such that a_i∈ {a, a + 2}

for all i ≥ 1 and ν^∗(α) = ν^∗([a]).

P r o o f. Let λ₁ = 1 and λ_2n+1 = n(λ₁+ λ₃+ . . . + λ_2n−1) for n ≥ 1.

Finally, put α = [0, a^λ1, a + 2, a^λ3, a + 2, a^λ5, . . .]. Then α is a U2-number according to Corollary 6 and ν^∗(α) = ν^∗([a]) by Theorem 5.1 in [3].

P r o o f o f T h e o r e m s 1 a n d 2. Let b be a positive even integer.

Then

[ν^∗([b]), ∞) = {ν^∗([b])} ∪ [∞ k=1

(ν^∗([b]), ν^∗([b + 2k]))

and both theorems follow from Lemmata 7 and 8, the theorem of Y. Dupain and V. T. S´os [6] and Theorem 3.1 of [3].

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Institut f¨ur Mathematik Universit¨at Wien Strudlhofgasse 4 A-1090 Wien, Austria

E-mail: baxa@pap.univie.ac.at

Received on 8.10.1996

and in revised form on 3.4.1997 (3057)