On the <span class="mathjax-formula">$b$</span>-ary expansion of an algebraic number

On the b-ary Expansion of an Algebraic Number.

YANNBUGEAUD

ABSTRACT- Letb2be an integer and denote byv(b) the number of its prime divisors. As a consequence of our main theorem, we establish that, fornlarge enough, there are at least (logn)1‡1=(v(b)‡4)(log logn) ¹⁼⁴ non-zero digits among the firstndigits of theb-ary expansion of every algebraic, irrational, real number. Analogous results are given for the Hensel expansion of an algebraic, irrationalp-adic number and for theb-expansion of1, whenbis either a Pisot, or a Salem number.

1. Introduction.

Let b2 be an integer. Despite some recent progress, theb-ary ex- pansion of an irrational algebraic number is still very mysterious. It is commonly believed that 

p2

, and every algebraic irrational number, should share mostof the properties satisfied by almostall real numbers (here and below, `almost all' always refers to the Lebesgue measure); in particular, every digit0;1;. . .;b 1 should appear in its b-ary expansion with the same frequency 1=b. However, it is still unknown whether, forb3, three different digits occur infinitely often in theb-ary expansion of 

p2 . There are several ways to measure the complexity of a real number. A first one is the block complexity. For a real numberuand a positive integer n, letdenote byp(n;u;b) the total number of distinct blocks ofndigits in theb-ary expansion ofu. It is shown in [2] with the help of a combinatorial transcendence criterion from [5] that the complexity functionn7!p(n;j;b) of an irrational algebraic numberjgrows faster than any linear function.

Since the complexity function of a rational number is uniformly bounded, this shows that the algebraic, irrational numbers are `not too simple' in this sense.

(*) Indirizzo dell'A.: Universite Louis Pasteur, U. F. R. de MatheÂmatiques, 7, rue Rene Descartes, 67084 Strasbourg, Cedex (France).

E-mail: bugeaud@math.u-strasbg.fr

Another point of view is taken in [6, 18], where the authors obtained very nice lower bounds for the number of occurrences of the digit 1 among the firstndigits of the binary expansion of an algebraic irrational number.

Apparently, their approach does not extend to a basebwithb3.

In the present paper, we consider the question of the complexity of an irrational algebraic number from a third point of view. Our aim is to es- timate the asymptotic behaviour of the number of digit changes in itsb-ary expansion. For a non-zero real numberu, write

uˆ X

k k0

a_k

b^kˆa _k0. . .a₀:a₁a₂. . .;

wherek₀0,a _k0 6ˆ0 ifk₀>0, thea_k's are integers fromf0;1;. . .;b 1g anda_kis non-zero for infinitely many indicesk. The sequence (a_k)_{k k0}is uniquely determined by u: itis its b-ary expansion. We then define the function nbdc, `number of digit changes', by

nbdc(n;u;b)ˆCardf1kn:a_k 6ˆa_k‡1g;

for any positive integern. Apparently, this function has not been studied up to now. If one believes that every irrational algebraic numberjbehaves in this respect like almost all numbers, then n7!nbdc(n;j;b) should grow linearly in terms ofn. Notsurprisingly, much less can be proved. Indeed, it easily follows from Ridout's Theorem [16], recalled in Section 3, that we have

nbdc(n;j;b)

logn !

n! ‡ 1 ‡ 1;

(1:1)

a proof of this is given at the beginning of Section 4. The main object of this note is to improve (1.1) by using a quantitative version of Ridout's Theorem.

As a corollary, we get lower and upper bounds for the sum of the firstn digits of an algebraic irrational number.

Our paper is organized as follows. The main results are stated in Sec- tion 2 and proved in Section 4, with the help of an auxiliary theorem given in Section 3. We display in Sections 5 and 6 the analogues of our main result for the Hensel expansion of irrational algebraicp-adic numbers and forb- expansions, respectively. Furthermore, we discuss in Section 7 various results on the transcendence of the seriesP

j1b ^nj, for an integerb2 and a rapidly growing sequence of positive integers (n_j)_j1. In the last section, we briefly consider an analogous problem for continued fraction expan- sions.

2. Results.

For any integerxat least equal to 2, we denote byv(x) the number of its prime divisors.

THEOREM1. Let b2be an integer. For every irrational, real alge- braic number j, there exists an effectively computable constant c(j;b), depending only onjand b, such that

nbdc(n;j;b)3 (logn)1‡1=(v(b)‡4)(log logn) ¹⁼⁴; (2:1)

for every integer nc(j;b).

We stress that for every non-zero rational numberp=q and for every integerb2, there exist integers`0 andCsuch that nbdc(n;p=q;b<sup>`</sup>)C for ``₀. The growth of the functionsn7!nbdc(n;u;b<sup>`</sup>) can be used to measure the complexity of the real numberu. In this respect, the `simplest' numbers are the rational numbers and Theorem 1 shows that algebraic irrational numbers are `not too simple'.

The proof of Theorem 1 given in Section 4 yields a (very) slightly better estimate than (2.1) and an explicit value for the constantc(j;b).

We display two immediate corollaries of Theorem 1. A first one con- cerns the number of non-zero digits in the expansion of a non-zero alge- braic number in an integer base, a question already investigated in [6, 18].

COROLLARY1. Let b2be an integer. Letjbe a non-zero algebraic number. Then, for n large enough, there are at least

(logn)1‡1=(v(b)‡4)(log logn) ¹⁼⁴

non-zero digits among the first n digits of the b-ary expansion ofj.

Forbˆ2, Corollary 1 gives a much weaker result than the one obtained by Bailey, Borwein, Crandall, and Pomerance [6], who proved that, among the first n digits of the binary expansion of a real irrational algebraic numberjof degreed, there are at leastc(j)n^1=doccurrences of the digit 1, wherec(j) is a suitable positive constant (see also Rivoal [18]).

Besides the number of non-zero digits, we may as well study the sum of digits. To this end, for an integerb2 and a non-zero real numberuwith b-ary expansion

uˆ X

k k0

a_k b^k;

set

S(n;u;b)ˆXⁿ

kˆ1

a_k;

for every positive integern. We know thatS(n;u;b)=ntends to (b 1)=2 withnfor almostall real numbersu. It is believed that the same holds when uis an arbitrary irrational algebraic number. In that case, we are only able to prove a much weaker result, namely thatn7! S(n;j;b) cannotincrease too slowly, nor too rapidly.

COROLLARY2. Let b2be an integer. Letjbe an irrational algebraic number. Then, for n large enough, we have

S(n;j;b)(logn)1‡1=(v(b)‡4)(log logn) ¹⁼⁴ (2:2)

and

S(n;j;b)n(b 1) (logn)1‡1=(v(b)‡4)(log logn) ¹⁼⁴:

Inequality (2.2) should be compared with Theorem 4.1 from [3], which gives an upper bound for the product of the firstnpartial quotients of an irrational, algebraic number. Actually, the proofs of both results are very similar.

3. An auxiliary result.

For a prime number ` and a non-zero rational number x, we set jxj<sub>`</sub>:ˆ` <sup>u</sup>, where uis the exponent of`in the prime decomposition ofx.

Furthermore, we set j0j_`ˆ0. With this notation, the main result of [16]

reads as follows.

THEOREM(Ridout, 1957). Let S₁and S₂be disjoint finite sets of prime numbers. Letube a real algebraic number. Letebe a positive real number.

Then there are only finitely many rational numbers p=q with q1such

that

u p q

Y

`2S1

jpj_`Y

`2S2

jqj_` < 1 q^2‡e (3:1)

For the proof of Theorem 1, we need an explicit upper bound for the number of solutions to (3.1). The best one for our purpose has been es- tablished by Locher [13].

We normalize absolute values and heights as follows. LetK be an al- gebraic number field of degreek. LetM(K) denote the set of places onK.

ForxinKand a placevinM(K), define the absolute valuejxj_vby (i) jxj_vˆ js(x)j^1=k ifvcorresponds to the embeddings:K,!R;

(ii) jxj_vˆ js(x)j^2=k ˆ js(x)j^2=k if v corresponds to the pair of con- jugate complex embeddingss;s:K,!C;

(iii) jxj_vˆ(Np) ^ordp(x)=kifvcorresponds to the prime idealpofO_K. Here,Npis the norm ofpand ord_p(x) is the exponent ofpin the de- composition of the ideal (x) into prime ideals. These absolute values satisfy the product formula

Y

v2M(K)

jxj_vˆ1 forxinK:

Theheightofxis then defined by H(x)ˆ Y

v2M(K)

maxf1;jxj_vg:

It does not depend on the choice of the number fieldKcontainingx. The Mahler measure ofxis given by

M(x)ˆH(x)^k0; wherek⁰kis the degree ofx.

We denote byj j_v an extension ofj j_vto the algebraic closure ofK.

THEOREM L. Let Kbe an algebraic number field. Let e be real with 0<e<1. LetF=Kbe an extension of number fields of degree f . Let S be a finite set of places onKof cardinality s. Suppose that for each v in S we are given a fixed elementuvinF. Let H be a real number with HH(uv)for all v in S. Consider the inequality

Y

v2S

minf1;ju_v gj_vg<H(g) ^{2 e} (3:2)

to be solved in elementsginK. Then there are at most e^7s‡19e ^{s 4}log (6f)log e ¹log (6f) solutionsginKto(3.2)with

H(g)maxfH;4^4=eg:

PROOF. This is Theorem 2 from [13] in the casedˆ1, with different

notation. p

4. Proof of Theorem 1.

Preliminaries.

Without any loss of generality, let us assume that (b 1)=b<j<1:

(4:1) Write

jˆX

k1

a_k

b^k ˆ0:a₁a₂. . .;

and define the increasing sequence of positive integers (nj)_j1 by a1ˆ. . .ˆan1,an1 6ˆan1‡1andanj‡1ˆ. . .ˆanj‡1,anj‡1 6ˆanj‡1‡1for every j1. Observe that

nbdc(n;j;b)ˆmaxfj:n_jng

fornn1, and that njj forj1. To construct good rational approx- imations tojwe simply truncate itsb-ary expansion atrankanj‡1and then complete with repeating the digitanj‡1. Precisely, forj1, we define the rational number

j_j:ˆX^nj

kˆ1

a_k b^k‡ X^{‡ 1}

kˆnj‡1

a_nj‡1 b^k ˆX^nj

kˆ1

a_k

b^k‡ a_nj‡1 b^nj(b 1) Set

P_j(X)ˆanj‡1 anj‡(anj anj 1)X‡. . .‡(a2 a1)X^{nj 1}‡a1X^nj

ˆa_nj‡1 a_nj‡(a_{nj 1‡1} a_{nj 1})X^{nj nj 1}‡. . .‡

(an1‡1 an1)X^{nj n1}‡a1X^nj; and observe that

j_j ˆ Pj(b) b^nj(b 1) (4:2)

Letp_jandq_jbe the coprime positive integers such that j_jˆp_j

q_j (4:3)

Since 1 janj‡1 anjj b 1, there exists a prime divisor`ofbsuch that ord`(anj‡1 anj)<ord`(b), thus

ord`

a_nj‡1 a_nj b^nj(b 1)

(nj 1)ord`(b) 1:

(4:4)

We infer from (4.4) and ord_`

(anj 1‡1 anj 1)b^{nj nj 1 1}‡. . .‡(a2 a1)b^{nj 2}‡a1b^{nj 1} b^{nj 1}(b 1)

(n_j 1) ord`(b) thatq_j 2^{nj 1}. Consequently, we get

2^{nj 1}H(j_j)ˆqjb^nj(b 1):

(4:5)

Observe that

jj j_jj< 1 b^nj‡1 (4:6)

Proof of inequality(1.1).

We first establish the result claimed in the introduction, that is, that Ridout's Theorem implies (1.1). Letebe a real number with 0<e<1. Letj be a positive integer. It follows from Ridout's Theorem that there exists a positive constantC(e) such that

j Pj(b) b^nj(b 1)

C(e)b ^nj(1‡e):

We then deduce from (4.6) that there is a positive constantC⁰(e) such that n_j‡1(1‡e)n_j‡C⁰(e) forj1. This implies thatn_j C⁰⁰(e)(1‡e)^jholds forj1 and some positive constantC⁰⁰(e). Consequently, we get

nbdc(n;j;b)logn

e 2 logC⁰⁰(e)

log (1‡e)logn 2e ; whennis sufficiently large. This proves (1.1).

The improvement upon (1.1) rests on the fact that, by Theorem L, we have an explicit estimate for the number of solutions to (3.1).

Preparation for the proof of Theorem1.

Letdenote bydthe degree ofj.

The Liouville inequality as stated by Waldschmidt [20], p. 84, asserts

that j p

q

1

M(j)(2q)^d; for any positive integersp;q:

Consequently, we obtain that

b^nj‡1(2b)^dM(j)b^dnj; thus we have

nj‡12dnj; (4:7)

for every integerjwithj2 log (3M(j)).

Let e be a positive real number with e<1=(log 6d). Let j₁ be the smallest integer with

j₁>max

2dlog 3M(j)

;10(2d‡1)e ¹logb ; (4:8)

and letjj1be an integer. Let`1;. . .; `_v(b) denote the (distinct) prime di- visors ofb. We infer from (4.6), (4.7), (4.2), (4.3) and the assumption (4.1) that

minf1;j1=j qj=pjjg Y^v(b)

iˆ1

jqjj_`i <4 minf1;jj pj=qjjg Y^v(b)

iˆ1

jqjj_`i

<4b ^nj‡1q_j¹(b 1)

4(b 1)^1‡2d (b 1)b^nj _nj‡1=nj q_j¹ 4(b 1)^1‡2dH(j_j) ^{1 nj‡1=nj}: (4:9)

Itfollows from (4.5) and (4.8) thatH(j_j)^e2^{ej 1}>4(b 1)^1‡2d. Conse- quently, we infer from (4.9) that, ifnj‡1(1‡2e)nj holds, then

minf1;j1=j q_j=p_jjg Y^v(b)

iˆ1

jq_jj_`i4(b 1)^1‡2dH(j_j) ^eH(j_j) ^{2 e}

<H(j_j) ^{2 e}: (4:10)

Observe that all the rational numbersj_jare distinct. Furthermore, we have H(j_j)maxfH(j);4^4=egas soon asjj₁.

We now apply Theorem L withKˆQ,Sˆ f1; `<sub>1</sub>;. . .; `_v(b)g,u₁ˆ1=j, u`iˆ0 foriˆ1;. . .;v(b). Letsˆ1‡v(b) be the cardinality ofSand set

T(e):ˆ2e^7s‡19e ^{s 4}(loge ¹) log (6d):

(4:11)

Itthen follows from (4.10), our choice ofe, and Theorem L that for every

integerjj1outside a set of cardinality at mostT(e) we have n_j‡1(1‡2e)n_j:

Completion of the proof of Theorem1.

LetJbe an integer with J>max

2dlog 3M(j)₂

;(4db)⁶ : Letj₂be the smallest integer with

j2>max

2dlog 3M(j)

;4db 

pJ

; and check thatJj³⁼²₂ .

Let k3 be an integer. It is convenient to set sˆs‡4ˆv(b)‡5.

Forhˆ1;. . .;k, define

e_hˆJ ^{(sk sh 1)=(sk‡1 1)}(logJ)^1=s(log 4d)^{(sk h‡1)=sk‡1h}; (4:12)

and check that

J ¹⁼⁴<e1<e2<. . .<ek<1=(log 6d):

Put S0 ˆ fj2;j2‡1;. . .;Jg. Observe that our choices of j2 and e1;. . .;e_k imply the analogue of (4.8), namely that

j2>max

2dlog 3M(j)

;10(2d‡1)e_h¹logb ; (1hk):

For hˆ1;. . .;k, let S_h denote the set of positive integers j such that

j2jJandnj‡1(1‡2eh)nj. Observe thatS0 S1. . . Sk. Itfol- lows from (4.10) that, for anyjinSh, the rational numberpj=qj satisfies

minf1;j1=j q_j=p_jjg Y^v(b)

iˆ1

jq_jj_`i<H(j_j) ^{2 eh}: Consequently, the cardinality ofS_his atmostT(e_h).

Write

S0ˆ(S0n S1)[(S1n S2)[. . .[(Sk 1n Sk)[ Sk: Combining (4.7) with the above estimates, we obtain that

nJ

n_j2ˆ nJ

n_{J 1}nJ 1

n_{J 2}. . .n_j2‡1 n_j2 (1‡2e1)^J Y^k

hˆ2

(1‡2e_h)^T(eh1)(2d)^T(ek):

Taking the logarithm and using the fact that log (1‡u)<ufor any posi- tive real numberu, we get

logn_J logn_j22Je₁‡2X^k

hˆ2

e_hT(e_{h 1})‡T(e_k)(log 2d):

(4:13)

We easily infer from (4.11), (4.12), and (4.13) that

lognJ lognj2 3ke^7v(b)‡26J^{(s 1)=s}J^{(s 1)=sk}(logJ)^1=s(log 4d)^1=s

‡J^{(s 1)=s}J^{(s 1)=sk}(logJ)^1=s(log 4d)^{(sk 1‡1)=sk}: (4:14)

Letlog^‡be the function defined on the positive real numbers by log^‡xˆ

ˆmaxflogx;1g. Choosing for k the smallest integer greater than (log logJ)(log^‡log log 4d), we getfrom (4.14) that

logn_J logn_j2J^{(s 1)=s}(logJ)^1=s(log logJ) (log 4d)^1=s(log^‡log log 4d);

where, as in the sequel of the proof, the constant implied bydepend only onv(b). Sincenj2 j2J²⁼³J^{(s 1)=s}, we getthat

logn_JJ^{(s 1)=s}(logJ)^1=s(log logJ) (log 4d)^1=s(log^‡log log 4d);

thus

J(logn_J)^{1‡1=(s 1)}(log logn_J) ^{1=(s 1)} (log log logn_J) ^{1 1=(s 1)}

(log 4d) ^{1=(s 1)}(log^‡log log 4d) ^{1 1=(s 1)}: Sinces 1ˆv(b)‡4>4, this completes the proof of Theorem 1. p

5. On the Hensel expansion of irrational algebraicp-adic numbers.

Letpbe a prime number. As usual, we denote byQ_pthe field ofp-adic numbers, and we call algebraic (resp. transcendental) any element ofQ_p which is algebraic (resp. transcendental) overQ. Ifj_pis ap-adic number, we denote by

j_pˆ X^‡1

kˆ k0

akp^k; (ak 2 f0;1;. . .;p 1g;k00;a k06ˆ0 if k0>0);

its Hensel expansion. It is proved in [2], Section 6, that the sequence (a_k)_{k k0} cannotbe `too simple' whenj_p is algebraic irrational. Using the methods developed in the present paper, we get thep-adic analogues of

Theorem 1 and Corollaries 1 and 2. With the above notation, for a positive integern, set

nbdc(n;j_p;p)ˆCardf1kn:a_k 6ˆa_k‡1g;

and

S(n;j_p;p)ˆXⁿ

kˆ1

a_k:

THEOREM2. Let p be a prime number. Letj_pbe an algebraic irrational number inQ_p. For any positive real numberdwithd<1=9and any suf- ficiently large integer n, we have

nbdc(n;j_p;p)(logn)^1‡d;

and there are at least(logn)^1‡dnon-zero digits among the first n digits of the Hensel expansion ofj_p, and, moreover,

(logn)^1‡d S(n;j_p;p)n(p 1) (logn)^1‡d:

The proof of Theorem 2 uses the same idea as that of Theorem 1. The good approximations toj_pare obtained by truncating its Hensel expansion and repeating the last digit. Theorem L, however, cannot be applied in the present context: we need a statement involving rational approximation to a p-adic algebraic number that gives an explicit upper bound for the number of solutions to an inequality first studied by Ridout [17]. A suitable statement is Theorem 3.1 from Evertse and Schlickewei [12], that we apply withnˆsˆ2 to get Theorem 2. We leave the details of the proof to the reader.

6. Onb-expansions.

Letb>1 be a real number. Theb-transformationTbis defined on [0;1]

byTb:x7 !bxmod 1. ReÂnyi [15] introduced theb-expansion of a realxin [0;1], denoted byd_b(x), and defined by

d_b(x)ˆ0:x₁x₂. . .x_n. . .;

where x_iˆ[bTⁱ_b ¹(x)]. For x<1, this expansion coincides with the re- presentation ofxcomputed by the `greedy algorithm'. Ifbis an integer, the digitsxiofxlie in the setf0;1;. . .;b 1ganddb(x) corresponds to theb- ary expansion ofxdefined in Section 1. Whenbis not an integer, the digits

xi lie in the setf0;1;. . .;[b]g. We direct the reader to [4] and to the re- ferences quoted therein for more on b-expansions. We stress that theb- expansion of 1 has been extensively studied.

It turns out that the method of proof of Theorem 1 applies tob-expan- sions, whenbis a Pisotor a Salem number. Recall thata Pisot(resp. Salem) number is a real algebraic integer>1, whose complex conjugates lie inside the open unit disc (resp. inside the closed unit disc, with at least one of them on the unit circle). In particular, every integerb2 is a Pisotnumber.

For a real algebraic numberu, letr(u) denote the sum of the number of infinite places on the number fieldQ(u) plus the number of distinct prime ideals inQ(u) that divide the norm ofu. In particular, r(b)ˆv(b)‡1 for every integerb2.

THEOREM3. Letb>1be a Pisot or a Salem number. Let M be a po- sitive integer and let(a_k)_k1 be a non-ultimately periodic sequence of in- tegers fromf0;1;. . .;Mg, and set

jˆX

k1

a_k b^k

Let d be a positive real number with d<1=(r(b)‡3). If there are arbi- trarily large integers n for which

Cardf1kn:ak6ˆak‡1g<(logn)^1‡d; thenjis transcendental.

We omit the proof of Theorem 3 since it is very similar to that of Theorem 1. Note that Ridout's Theorem has already been used in this context, see [1].

We display a consequence of Theorem 3 on the number on non-zero digits in theb-expansion of 1.

COROLLARY3. Letb>1be a Pisot or a Salem number. Write db(1)ˆ0:a1a2. . .

For any positive real numberdwithd<1=(r(b)‡3)and any sufficiently large integer n, we have

a₁‡a₂‡. . .‡a_n>(logn)^1‡d;

and there are at least(logn)^1‡dindices j with1jn and a_j6ˆ0.

7. On the seriesP

j1 b ^nj.

Let b be an integer with b2. Let nˆ(n_j)_j1 be a non-decreasing sequence of positive integers. It is well-known that ifngrows sufficiently rapidly, then the number

j_n;bˆX

j1

b ^nj

is transcendental. For example, it easily follows from Ridout's Theorem that the assumption

lim sup

j! ‡ 1

n_j‡1 n_j >1

implies the transcendence ofj_n;b, see e.g. Satz 7 from the monograph [19].

In particular, for any positive real number e, the real number j_n;b is transcendental when njˆ2^[ej], where [ ] denotes the integer part. A much sharper statement follows from Theorem 1.

COROLLARY4. Let b2be an integer. Lethbe a real number with 0h< 1

v(b)‡5: Then, the sum of the series

X

j1

b ^nj; where n_jˆ2^[j1h] for j1;

is transcendental.

PROOF. Letybe a real number with 0y<1. LetNbe a large integer.

We check that the number of positive integersjsuch that 2^[j1y]Nis less than some absolute constant times (logN)^{1=(1 y)}, and we apply Theorem 1 to

conclude. p

Actually, it is possible to derive from Theorem 1 the transcendence of the real numberj_n;bfor other fast growing sequencesn. Like in Corollary 4, the precise statement would involvev(b), the number of distinct prime factors ofb. It turns out that it is possible to improve Corollary 4 and to get rid of the dependence on v(b) by using the following version of the Cu- giani-Mahler Theorem [10, 11, 14], that we extract from Bombieri and Gubler [7] (see also Bombieri and van der Poorten [8]).

THEOREMBG. Let S1 and S2be disjoint finite sets of prime numbers.

Letube a real algebraic number of degree d. For any positive real number t set

f(t)ˆ7 (log 4d)¹⁼²

log log (t‡log 4) log (t‡log 4)

₁₌₄ Let(p_j=q_j)_j1be the sequence of rational solutions of

u p q

Y

`2S1

jpj_`Y

`2S2

jqj_` 1 q^2‡f(logq);

ordered such that1q1<q2<. . .Then either the sequence(p_j=q_j)_j1is finite or

lim sup

j! ‡ 1

logq_j‡1

logqj ˆ ‡ 1:

PROOF. This follows from Theorem 6.5.10 of [7]. p We can then proceed exactly as Mahler did ([14], Theorem 3, page 178).

THEOREM4. Let b2 be an integer. Let(a_j)_j1 be a non-decreasing sequence of positive integers tending to infinity. Letnˆ(n_j)_j1be an in- creasing sequence of positive integers satisfying n₁3and

n_j‡1

1‡an

log logn_j logn_j

₁₌₄

n_j; (j1):

Then the real number

j_n;bˆX

j1

b ^nj

is transcendental.

We omit the proof since it is exactly the same as Mahler's one. Note that Theorem 1 yields a similar statement as Theorem 4, with however 1=4 replaced by any real number smaller than 1=(v(b)‡4).

It does not seem to us that Theorem 1 can be derived from Theorem BG.

An easy computation allows us to deduce from Theorem 4 the following improvementof Corollary 4.

COROLLARY5. Let b2be an integer. Lethbe a real number with 0h<1

5: Then, the sum of the series

X

j1

b ^nj; where njˆ2^[j1h] for j1;

is transcendental.

More generally, one can consider the following question.

PROBLEM. Letnˆ(nj)j1 be a strictly increasing sequence of positive integers and set

fn(z)ˆX

j1

z^nj: (7:1)

If the sequencenincreases sufficiently rapidly, then the function fntakes transcendental values at every non-zero algebraic point in the open unit disc.

By a clever use of the Schmidt Subspace Theorem, Corvaja and Zannier [9] proved that the conclusion of the Problem holds for fn given by (7.1) when the strictly increasing sequencensatisfies

lim inf

j! ‡ 1

nj‡1

n_j >1:

(7:2)

It is perhaps possible to combine the method developed in [9] with the ideas of the present paper to relax the assumption (7.2).

8. Continued fraction expansions.

Despite some recent progress, the continued fraction expansion of a real algebraic number of degree greater than or equal to three remains very mysterious. In particular, we still do not know whether its sequence of partial quotients is bounded or not. Even an analogue of Theorem 1 for continued fraction expansions would be new.

In the sequel, we restrict our attention to expansions with partial quotients equal to 1 or 2. More precisely, for a strictly increasing sequence

nˆ(nj)_j1 of positive integers, we denote byj_n :ˆ[0;a1;a2;. . .] the real number whose continued fraction expansion is given, fork1, byakˆ1 if k2=nanda_kˆ2 ifk2n.

It is expected that j_n is transcendental unless (a_k)_k1 is ultimately periodic (in which casej_nis clearly a quadratic number). In particular, it is very likely thatj_nis transcendental ifnincreases sufficiently rapidly. As a first result in this direction, we claim that, ifj_n is algebraic, then

logn_jˆo(j):

(8:1)

Indeed, if (8.1) does not hold, then there are infinitely many positive in- tegersjand a real numberc>1 such thatnj>c^j. Consequently, we have

lim sup

j! ‡ 1

n_j‡1 n_j >1;

and the transcendence of the associated real number j_n follows from Theorem 3.2 of [3].

With similar ideas as in the present paper, but with some additional technical difficulties due to the use of the Schmidt Subspace Theorem in place of the Ridout Theorem, we are able to improve the above result as follows:

There exists a positive real numberdsuch that, ifj_nis algebraic, then logn_jj^{1 d}

holds for every sufficiently large integer j.

This and related results will be the subject of a forthcoming paper.

Acknowledgements. I am very pleased to thank the referee for having written quickly a detailed report that helped me to improve the pre- sentation of the paper.

REFERENCES

[1] B. ADAMCZEWSKI,Transcendance «aÁ la Liouville» de certains nombres reÂels, C. R. Acad. Sci. Paris,338(2004), pp. 511-514.

[2] B. ADAMCZEWSKI- Y. BUGEAUD,On the complexity of algebraic numbers I.

Expansions in integer bases, Ann. of Math.,165(2007), pp. 547-565.

[3] B. ADAMCZEWSKI- Y. BUGEAUD,On the Maillet-Baker continued fractions, J.

reine angew. Math.,606(2007), pp. 105-121.

[4] B. ADAMCZEWSKI - Y. BUGEAUD, Dynamics for b-shifts and Diophantine approximation, Ergodic Theory Dynam. Systems. To appear.

[5] B. ADAMCZEWSKI- Y. BUGEAUD- F. LUCA,Sur la complexite des nombres algeÂbriques, C. R. Acad. Sci. Paris,339(2004), pp. 11-14.

[6] D. H. BAILEY- J. M. BORWEIN- R. E. CRANDALL- C. POMERANCE, On the binary expansions of algebraic numbers, J. TheÂor. Nombres Bordeaux,16 (2004), pp. 487-518.

[7] E. BOMBIERI- W. GUBLER,Heights in Diophantine Geometry. New mathe- matical monographs 4, Cambridge University Press, 2006.

[8] E. BOMBIERI- A. J.VAN DERPOORTEN,Some quantitative results related to Roth's theorem, J. Austral. Math. Soc. Ser. A,45(1988), pp. 233-248.

[9] P. CORVAJA- U. ZANNIER,Some new applications of the subspace theorem, Compositio Math.,131(2002), pp. 319-340.

[10] M. CUGIANI,Sull'approssimabilitaÁ di un numero algebrico mediante numeri algebrici di un corpo assegnato, Boll. Un. Mat. Ital.,14(1959), pp. 151-162.

[11] M. CUGIANI,Sulla approssimabilitaÁ dei numeri algebrici mediante numeri razionali, Ann. Mat. Pura Appl.,48(1959), pp. 135-145.

[12] J.-H. EVERTSE- H. P. SCHLICKEWEI,A quantitative version of the absolute subspace theorem, J. reine angew. Math.,548(2002), pp. 21-127.

[13] H. LOCHER,On the number of good approximations of algebraic numbers by algebraic numbers of bounded degree, Acta Arith.,89(1999), pp. 97-122.

[14] K. MAHLER,Lectures on Diophantine approximation, Part 1: g-adic numbers and Roth's theorem, University of Notre Dame, Ann Arbor, 1961.

[15] A. REÂNYI, Representations for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hung.,8(1957), pp. 477-493.

[16] D. RIDOUT,Rational approximations to algebraic numbers, Mathematika,4 (1957), pp. 125-131.

[17] D. RIDOUT, The p-adic generalization of the Thue-Siegel-Roth theorem, Mathematika,5(1958), pp. 40-48.

[18] T. RIVOAL,On the bits couting function of real numbers. Preprintavailable at:

http://www-fourier.ujf-grenoble.fr/ rivoal/articles.html

[19] T. SCHNEIDER,EinfuÈhrung in die transzendenten Zahlen. Springer-Verlag, Berlin-GoÈttingen-Heidelberg, 1957.

[20] M. WALDSCHMIDT,Diophantine Approximation on Linear Algebraic Groups.

Transcendence properties of the exponential function in several variables.

Grundlehren der Mathematischen Wissenschaften 326. Springer-Verlag, Berlin, 2000.

Manoscritto pervenuto in redazione il 13 settembre 2006.