Algebraic independence of the values of power series with unbounded coefficients

c 2017 by Institut Mittag-Leffler. All rights reserved

Algebraic independence of the values of power series with unbounded coefficients

Kaneko Hajime

Abstract. Many mathematicians have studied the algebraic independence overQof the values of gap series, and the values of lacunary series satisfying functional equations of Mahler type. In this paper, we give a new criterion for the algebraic independence overQof the values

_∞

n=0t(n)β⁻ⁿ for distinct sequences (t(n))^∞_n=0of nonnegative integers, whereβis a fixed Pisot or Salem number. Our criterion is applicable to certain power series which are not lacunary.

Moreover, our criterion does not use functional equations. Consequently, we deduce the algebraic independence of certain values_∞

n=0t1(n)β⁻ⁿ, ...,_∞

n=0tr(n)β⁻ⁿsatisfying lim

n→∞,t_i−1(n)=0

ti(n)

ti−1(n)^M =∞ (i= 2, ..., r) for any positive real numberM.

1. The transcendence of the values of power series with bounded coefficients

We introduce notation which we use throughout this paper. LetN(resp. Z⁺) be the set of nonnegative integers (resp. positive integers). For a real numberx, we denote the integral and fractional parts of xbyxand {x}, respectively. We use the Landau symbolso, O, and the Vinogradov symbols,with their regular meanings. For a sequence of integerst=(tn)^∞_n=0, put

S(t) :={n∈N|tn= 0}

and

f(t;X) :=

∞ n=0

t_nXⁿ. (1.1)

Key words and phrases:algebraic independence, Pisot numbers, Salem numbers.

2010 Mathematics Subject Classification:primary 11J99; secondary 11K16, 11K60.

Note that iftn≥0 for anynand iftn∈{0,1}for any sufficiently largen, then (1.1) is rewritten as

f(t;X) = ∞ m=0

X^vm, (1.2)

where (v_m)^∞_m=0 is a sequence of nonnegative integers satisfying v_m+1>v_m for any sufficiently largem. LetAbe a nonempty subset ofN. Set

λ(A;R) := Card{[0, R)∩A}

(1.3)

for anyR≥1, where Card denotes the cardinality.

In this paper, we study arithmetical properties of the valuesx=f(t;α) for a fixed algebraic numberαwith 0<|α|<1. In this section, we review known results of the transcendence of such values in the case where the coefficients are bounded.

In particular, ifαis a real number with 0<α<1, putβ:=α⁻¹. In this paper, an infinite series of the form

x=f(t;β⁻¹) = ∞ n=0

tnβ⁻ⁿ,

where t=(t_n)^∞_n=0 is a sequence of integers, is called a β-representation ofx. The β-expansion ofx, introduced by Rényi [17], is aβ-representation computed by the greedy algorithm. LetTβ:[0,1)→[0,1) be theβ-transformation defined byTβ(x):=

{βx} forx∈[0,1). Then theβ-expansion ofx∈[0,1) is denoted by

x= ∞ n=1

tn(β;x)β⁻ⁿ,

wheretn(β;x)=βT_βⁿ⁻¹(x)forn≥1.

In the rest of this section, v=(vm)^∞_m=0 denotes an ultimately increasing se- quence of nonnegative integers. Recall that β >1 is a Pisot number if β is an algebraic integer whose conjugates except itself have absolute value less than 1. In particular, any integer greater than 1 is a Pisot number. Moreover, an algebraic integer β >1 is a Salem number if the conjugates of β except itself have absolute values not greater than 1 and ifβ has at least one conjugate with absolute value 1.

Adamczewski [1] showed that if lim sup

m→∞

v_m+1 vm

>1,

then_∞

m=0β^−vm is transcendental for any Pisot or Salem number β. In the case where α is a general algebraic number with 0<|α|<1, Corvaja and Zannier [7]

showed that if

lim inf

m→∞

vm+1

v_m >1, then_∞

m=0α^vm is transcendental.

However, it is difficult to study the transcendence in the case where

mlim→∞

vm+1

v_m = 1.

(1.4)

We review known results on the transcendence of certain values _∞

m=0β^−vm sat- isfying (1.4) in the case where β is a Pisot or Salem numbers. The results are obtained by using the partial results on the normality of theβ-expansions of alge- braic irrational numbers. For instance, consider the case where β=b is an integer greater than 1. Then Borel [4] conjectured that all algebraic irrational numbers are normal in base-b, which is still open. If Borel’s conjecture is true, then we have the following: if

lim sup

m→∞

v_m m =∞, then_∞

m=0b^−vm is transcendental.

Let x∈[0,1) be an algebraic irrational number. For a positive integer N, let μ(b, x;N) be the number of nonzero digits in the firstN digits ofxin base-b. In the case of b=2, Bailey, Borwein, Crandall, and Pomerance [3] gave lower bounds for μ(2, x;N) withN≥N0, whereN0is an ineffective positive constant. Modifying the proof of the above results, Adamczewski, Faverjon [2], and Bugeaud [5] indepen- dently gave effective versions of the lower bounds forμ(b, x;N) with general integer b≥2. Lower bounds for the numbers of nonzero digits in β-representations of real numbers were also studied [10] and [11] in the case where β is a Pisot or Salem number.

Using the lower bounds, we deduce for each Pisot or Salem number β that if v=(v_m)^∞_m=0 satisfies

lim sup

m→∞

vm

m^A=∞ (1.5)

for any positive real number A, then _∞

m=0β^−vm is transcendental (see Corol- lary 2.4 in [10]). Ifβ=b>1 is an integer, then the transcendence of_∞

m=0b^−vm was essentially proved by Bailey, Borwein, Crandall, and Pomerance [3]. Consequently, we obtain the transcendence of_∞

m=0β^−vm for certain classes of_∞

m=0β^−vm sat- isfying (1.4).

For instance, put, for any positive realν, σ_1,ν(m) := exp

(logm)^1+ν

=m^(logm)ν (m= 1,2, ...), and let

σ2(m) := exp(logmlog logm) =m^{log logm} (m= 3,4, ...).

For any Pisot or Salem numberβ, we obtain the transcendence of ∞

m=1

β^−σ1,ν(m) and ∞ m=3

β^−σ2(m) (1.6)

because (σ1,ν(m))^∞_m=1 and (σ2(m))^∞_m=3 fulfill (1.5).

In what follows, we study the algebraic independence overQof valuesf(t;α) for a fixed algebraic numberαwith 0<|α|<1 and distinct sequencest. We also in- vestigate the case wheretis unbounded. In Section2, we introduce known results on the algebraic independence of the values of power series satisfying certain lacunary assumption. In Section3, we give main criterion for the algebraic independence in the case ofα=β⁻¹, where β is a Pisot or Salem number. Using our criterion, we deduce the algebraic independence of real numbers including the values (1.6), which gives new examples of algebraic independence. We prove our criterion in Section4.

2. Algebraic independence of the values of lacunary series

Let t=(tn)^∞_n=0 be a sequence of integers such that tn=0 for infinitely many n’s. Put

{n∈N|t_n= 0}=:{w_t(0)< w_t(1)< ...}. We say thatf(t;X) is a gap series if

mlim→∞

wt(m+1) w_t(m) =∞. For instance,_∞

m=0X^m! is a gap series. Moreover, we say thatf(t;X) is lacunary if

lim inf

m→∞

w_t(m+1) wt(m) >1.

For instance,

ϕ_k(X) :=

∞ m=0

X^km

is lacunary for any integerk≥2. In this section we introduce known results on the algebraic independence of the values of gap series, and lacunary series satisfying functional equations of Mahler type, respectively. In the rest of this section, α denotes an algebraic number with 0<|α|<1.

We consider the algebraic independence of the values of gap series. We first study the case wheref(t;X) has the form (1.2). Shiokawa [19] gave a criterion for the algebraic independence of gap series at algebraic points. Using his criterion, we obtain for eachαthat the set

_∞

m=0

α^(km)!

k= 1,2, ...

is algebraically independent, which generalizes the result by Schmidt [18].

Next we consider the case where t=(t_n)^∞_n=0 is unbounded. In particular, we study the algebraic independence of the set

f^(l)(t;α)l= 0,1, ...

,

wheref^(l)(t;X) denotes thel-th derivative off(t;X). For anym∈N, let Am:= max{1,|tn| |0≤n≤wt(m)}

and letRbe the convergence radius with 0<R≤1. Then Cijsouw and Tijdeman [6]

showed for anyαwith|α|<Rthat if

m→∞lim

w_t(m+1) wt(m)+logAm

=∞, (2.1)

thenf(t;α) is transcendental. Nishioka [15] showed for eachαwith|α|<Rthat if (2.1) holds, then the set

f^(l)(t;α)l= 0,1, ...

is algebraically independent.

Next, we investigate the algebraic independence of the values of lacunary series satisfying functional equations of Mahler type. For instance, ifkis an integer greater than 1, then the seriesϕk(X) satisfy the following:

ϕk(X^k) = ∞ m=1

X^km= ∞ m=0

X^km−X=ϕk(X)−X.

Using the above relation, Mahler [12] showed the transcendence ofϕ_k(α) for anyα.

In the rest of this section we introduce results on algebraic independence proved by Mahler’s method.

Nishioka [13] proved for anyαthat the set {ϕ_k(α)|k= 2,3, ...}

is algebraically independent. More generally, Tanaka [20] showed for any positive real numbersw₁, ..., w_r linearly independent overQand anyαthat the set

_∞

m=0

α^wikm

i= 1, ..., r, k= 2,3, ...

is algebraically independent.

We now consider the case wheret=(t_n)^∞_n=0 is unbounded. We call a sequence (w_n)^∞_n=0of integers a linear recurrence if there existr≥1 andc₁, ..., c_r∈Cwithc_r=0 such that, for anyn∈N,

w_n+r=c₁w_n+r−1+c₂w_n+r−2+...+c_rw_n. Nishioka [16] verified for any integerk≥2 and anyαthat the set

ϕ^(l)_k (α)l= 0,1, ...

is algebraically independent, using her criterion for algebraic independence in [14].

Considering the operatorX(d/dX), we see that the set _∞

m=0

k^lmα^km

l= 0,1, ...

is also algebraically independent. More generally, letrbe a positive integer and let s⁽¹⁾=(s1(m))^∞_m=0, ...,s^(r)=(sr(m))^∞_m=0 be linearly independent linear recurrences of integers. Using the same criterion in [14] as above, Nishioka [16] showed for any integerk≥2 and anyαthat the numbers

∞ m=0

si(m)α^km (i= 1, ..., r) are algebraically independent.

Tanaka [21] studied necessary and sufficient conditions for the algebraic inde- pendence of the values of_∞

m=0X^vm and their derivatives, wherev=(v_m)^∞_m=0 are distinct linear recurrences satisfying certain assumptions. For instance, letFm(m=

0,1, ...) be the Fibonacci sequence defined byF0=0, F1=1, andFm+2=Fm+1+Fm

for anym=0,1, .... Put

ϕ_F(X) :=

∞ m=0

X^Fm.

Then, for anyα, the set

ϕ^(l)_F (α)l= 0,1, ...

is algebraically independent.

3. Main results

Let β be a Pisot or Salem number and t=(t_n)^∞_n=0 a sequence of nonnegative integers such thatt_n=0 for infinitely manyn’s. In this section we study criteria for the algebraic independence off(t;β⁻¹) in the case wheref(t;X) is not generally a lacunary series. Note that we do not use functional equations for the criteria.

LetAbe a nonempty subset ofNandk∈N. Then set kA:=

{0} (ifk=0),

{s1+...+s_k|s1, ..., s_k∈A} (ifk≥1).

Recall thatλ(A;R) is defined by (1.3). Moreover, (σ1,ν(m))^∞_m=1 for a positive real numberν and (σ₂(m))^∞_m=3 are defined in Section1.

We first consider the case where t=(tn)^∞_n=0 is bounded. Using criteria for algebraic independence in [9], we obtain the following:

Theorem 3.1. ([9]) Let β be a Pisot or Salem number.

(1)The continuum set _∞

m=1

β^−σ1,ν(m)

ν∈R, ν≥1

is algebraically independent.

(2)Let ν andη be distinct positive real numbers. Then the two numbers ∞

m=1

β^−σ1,ν(m) and ∞ m=1

β^−σ1,η(m)

are algebraically independent.

(3)Let ν be a positive real number. Then the two numbers ∞

m=1

β^−σ1,ν(m) and ∞ m=3

β^−σ2(m)

are algebraically independent.

Note that ifβ=b is an integer greater than 1, then the first and second state- ments of Theorem3.1were obtained in [8].

In what follows, we introduce our main results. We now give a new criterion for algebraic independence applicable to some special cases, where t=(tn)^∞_n=0 is unbounded.

Theorem 3.2. Let Abe a set of nonnegative integers satisfying the following two assumptions:

1.There exists a positive constantC1>1such that [R, C1R)∩A =∅ (3.1)

for any sufficiently largeR.

2.For an arbitrary positiveε, lim inf

R→∞

λ(A;R) R^ε = 0.

(3.2)

Let t⁽¹⁾=(t1(n))^∞_n=0, ...,t^(r)=(tr(n))^∞_n=0 be sequences of nonnegative integers with S(t⁽¹⁾) =...=S(t^(r)) =A

(3.3)

satisfying the following two assumptions:

1.There exists a constant0<C2<1such that,for each i=1, ..., r, log⁺t_i(n) =o

n^1−C2 (3.4)

asntends to infinity, wherelog⁺x=log max{1, x} for a real number x.

2.LetM be an arbitrary positive real number. Then,fori=2, ..., r,

n∈A,n→∞lim ti(n)

t_i−1(n)^M =∞.

(3.5)

Let β be a Pisot or Salem number. Then the numbers f(t⁽¹⁾;β⁻¹), ..., f(t^(r);β⁻¹) are algebraically independent.

Theorem 3.2 implies that if a set A of nonnegative integers satisfying (3.1) and (3.2) is given, then we can deduce examples of algebraically independent real numbers, changing sequencest⁽¹⁾, ...,t^(r)of coefficients. Theorem 3.2is applicable even ift⁽¹⁾, ...,t^(r) are not linear recurrences.

We give examples of coefficients for a fixed infinite setAof nonnegative integers.

Lett=(t_n)^∞_n=0 be a sequence of integers. Put f_A(t;X) =

n∈A

t_nXⁿ.

Let Θ be the subset ofR²defined by

Θ :={(μ, ν)∈R²|0< μ <1}∪{(0, ν)∈R²|ν≥0}.

Moreover, for any (μ, ν)∈Θ, we define the sequencer_μ,ν=(r_μ,ν(n))^∞_n=0 by rμ,ν(n) :=

exp

(n+2)^μ

log(n+2)ν .

Corollary 3.3. Let A be a subset of Nsatisfying (3.1)and (3.2). Then, for any Pisot or Salem numberβ, the continuum set

G(A) :=

f_A(r_μ,ν;β⁻¹)|(μ, ν)∈Θ is algebraically independent.

Corollary3.3is easily seen by Theorem3.2. In fact, any elementf_A(r_μ,ν;β⁻¹) ofG(A) satisfies (3.4) byμ<1. Moreover, let f_A(rμ,ν;β⁻¹) and f_A(rμ,ν;β⁻¹) be any distinct two elements ofG(A). Suppose thatμ>μ, orμ=μ andν >ν. Then we see for any positive real numberM that

n→∞lim

r_μ,ν(n)

rμ,ν(n)^M = lim

n→∞

exp

(n+2)^μ

log(n+2)ν exp

M(n+2)^μ

log(n+2)ν=∞, which implies that (3.5) is also fulfilled.

Various sets A of nonnegative integers satisfy (3.1) and (3.2). Consider the case whereAis denoted as

A={v_m|m≥m₀}, (3.6)

where m0 is a nonnegative integer and (vm)^∞_m=m0 is an ultimately increasing se- quence of nonnegative integers. We see that if (v_m)^∞_m=m0 satisfies (1.5) and

lim sup

m→∞

vm+1

v_m <∞, (3.7)

then the setAdefined by (3.6) fulfills (3.1) and (3.2). In fact, letεbe an arbitrary positive real number. Using (1.5) withA=1/ε, we see for any positive real number ythat there exist infinitely many positive integersmwithvm>y^1/εm^1/ε. Thus, we obtain that

λ(A;y^1/εm^1/ε)

(y^1/εm^1/ε)^ε =Card([0, y^1/εm^1/ε)∩A)

ym ≤1

y and that

lim inf

R→∞

λ(A;R) R^ε ≤1

y,

which implies (3.2) becausey is an arbitrary positive real number.

Recall that vm:=σ1,ν(m) (m≥1) satisfies (1.5) and (1.4) for any positive real numberν. Thus,A1:={σ_1,ν(m)|m≥1} satisfies (3.1) and (3.2). Hence, A1 is applicable to Theorem3.2or Corollary3.3. In particular, Corollary3.3implies that

G(A1) :=

f_A1(rμ,ν;β⁻¹)|(μ, ν)∈Θ

is algebraically independent. Note for any nonnegative integer n that r_0,0(n)=2.

Thus, we see

1

2f_A1(r0,0;β⁻¹)−^∞

m=1

β^−σ1,ν(m)∈Q(β)

because (σ1,ν(m))^∞_m=1is ultimately increasing. Therefore, we deduce the algebraic independence of real numbers including the value

∞ m=1

β^−σ1,ν(m) Similarly,{σ₂(m)|m≥3}also fulfills (3.1) and (3.2).

Moreover, we note that if (vm)^∞_m=m0 satisfies 1<lim inf

m→∞

vm+1

v_m ≤lim sup

m→∞

vm+1

v_m <∞,

then (1.5) and (3.7) are satisfied. In particular, the set {wη^m|m≥0} for real numbers w>0, η >1, and the set{F_m|m≥0} for the Fibonacci sequence (F_m)^∞_m=0 fulfill (3.1) and (3.2).

In the last of this section we also deduce the algebraic independence of certain two numbers related to the derivatives of functions as follows:

Corollary 3.4. Let A be a subset of N satisfying (3.1) and (3.2). Let t=

(t(n))^∞_n=0 be a bounded sequence of positive integers. Let β be a Pisot or Salem number. Then, for any positive integerl, the numbersf_A(t;β⁻¹) andf_A^(l)(t;β⁻¹) are algebraically independent.

Proof. Letγ1:=f_A(t;β⁻¹) andγ2:=β^−lf_A^(l)(t;β⁻¹)+

n<l,n∈Aβ⁻ⁿ, where γ₂=

∞ n=l

n(n−1)...(n−l+1)t_nβ⁻ⁿ+

n<l,n∈A

β⁻ⁿ

=:

∞ n=0

t2(n)β⁻ⁿ.

Then two sequencest⁽¹⁾:=(t(n))^∞_n=0 and t⁽²⁾:=(t₂(n))^∞_n=0 satisfy (3.3), (3.4), and (3.5) becauset⁽¹⁾ is bounded. Thus, we obtain from Theorem 3.2that γ1 andγ2

are algebraically independent, which implies the corollary.

4. Proof of Theorem 3.2

We introduce notation. Letk be a positive integer andm=(m1, ..., mk)∈N^k, X:=(X₁, ..., X_k). Put

|m|:=m1+...+mk and X^m:=X₁^m1...X_k^mk. For convenience, ifk=0, then set

|m|:= 0 and X^m:= 1.

We write by grl the graded reverse lexicographical order on N^r as follows: Let k=(k₁, ..., k_r),k=(k₁, ..., k_r) be distinct elements of N^r. Thenkgrlk if and only if|k|>|k|, or|k|=|k|andkh>k_h, where

h= max{1≤i≤r|k_i=k_i}. In particular, we have

(0, ...,0,1)grl(0, ...,0,1,0)grl...grl(1,0, ...).

Putfi(X):=f(t⁽ⁱ⁾;X) andξi:=fi(β⁻¹) fori=1, ..., r. Set0:=(0, ...,0)∈N^randξ:=

(ξ1, ..., ξr). In what follows, we show thatP(ξ)=0 for any nonzero polynomialP(X) whose coefficients are integers. LetD be the total degree of P(X). Throughout the proof of Theorem3.2, we consider the set (D−1)B, where B is defined later.

However, (D−1)Bis defined only if D−1≥1. For simplicity, we may assume that D≥2. In fact, ifD=1, then it suffices to show that the polynomial X1P(X) does not vanish atX=ξ. Let

P(X) =:A0+

k∈Λ

AkX^k,

where Λ is a nonempty subset ofN^r\{0},A₀∈Z, andA_k∈Z\{0}for anyk∈Λ. We write byg=(g1, ..., gr) the maximal element of Λ with respect togrl. Without loss of generality, we may assume thatAg>0. Set

Λ1:={k∈Λ\{g} | |k|=D} and Λ2:={k∈Λ\{g} | |k|< D}.

Then we have

Λ ={g}∪Λ1∪Λ2.

In what follows, the implied constants in the symbolsandandC3, C4, ...are positive constants depending only onf₁(X), ..., f_r(X), β, andP(X). Set

C3:= max

⎧⎨

⎩1,2(D!) Ag

⎛

⎝1+

k∈Λ1

|Ak|

⎞

⎠

⎫⎬

⎭.

By (3.5), there exists a positive integerC4 such that if nis an element ofA with n≥C₄, then

tu(n)> C₃^Dtv(n)^D (4.1)

for any integers u, v with 1≤v <u≤r. For simplicity, we considerηi=ϕi(β⁻¹) in- stead ofξi fori=1, ..., r, where

ϕi(X) = ∞ n=0

si(n)Xⁿ:= 1+

∞ n=C4

ti(n)Xⁿ. (4.2)

Then we seeβ^C4(ξi−ηi)∈Z[β]. Observe for eachk=(k1, ..., kr)∈Λ that β^C4DA_kξ^k=β^C4(D−|k|)A_k

r i=1

β^C4(ξ_i−η_i)+β^C4η_iki

is a polynomial ofη=(η1, ..., ηr) whose coefficients are elements ofZ[β]. Expanding β^C4DP(ξ) by the relation above, we see that

β^C4DP(ξ) =:Q(η) =B₀+

k∈Γ

B_kη^k, (4.3)

where Q(X)∈Z[β][X1, ..., Xr] has total degree D(≥2), Γ is a nonempty subset of N^r\{0}, B₀∈Z[β],andB_k∈Z[β]\{0}for any k∈Γ. By the definition ofQ(X), the maximal element of Γ with respect togrlisg. Similarly, putting

Γ₁:={k∈Γ\{g} | |k|=D} and Γ₂:={k∈Γ\{g} | |k|< D}, we get Γ1=Λ1and

Bk=β^C4DAk

(4.4)

for anyk∈{g}∪Γ₁. In particular, we haveB_g>0. Note that Γ ={g}∪Γ₁∪Γ2.

We check that the power seriesϕ1(X), ..., ϕr(X) with nonnegative integral coeffi- cients satisfy the assumptions of Theorem3.2. In particular, we get

S(ϕ₁) =...=S(ϕ_r)

={0}∪([C₄,∞)∩A) =:B by (3.3), (4.2), and so

[R, C₁R)∩B =∅ (4.5)

for any sufficiently largeR by (3.1). Moreover, putting λ(B;R) =:λ(R) and C:=C2/2 for simplicity, we see for an arbitrary positive realεthat

lim inf

R→∞

λ(R) R^ε = 0

by (3.2). By considering the case ofε=C/(2D−1), there exists an infinite setF of nonnegative integers such that

4^{−1/(2D−1)}(1+C1)^{−1/(2D−1)}> λ(N) N^C/(2D−1) for anyN∈F. Thus, we get for anyN∈F that

N^C>4(1+C₁)λ(N)^2D−1. (4.6)

We see by (3.4) that

log⁺s_i(n) =o n^1−2C (4.7)

asntends to infinity, where 0<2C<1. Moreover, using (4.1) and (4.2), we get su(n)> C₃^Dsv(n)^D

(4.8)

for any integersu, v with 1≤v <u≤rand anyn∈B\{0}.

We now calculateη^kfork∈Γ. Letkbe a positive integer andm=(m1, ..., mk)∈

N^k. Fori=1, ..., r, put

s_i(m) :=s_i(m₁)...s_i(m_k).

(4.9)

For convenience, ifk=0, then we setsi(m):=1. For a positive integerm, we denote by B^m the mth Cartesian power ofB. Moreover, putB⁰:={0}. For any η^k with k=(k1, ..., kr)∈Γ, we see

η^k= r i=1

m∈B

si(m)β^−m ki

= r i=1

⎛

⎝

mi∈B^ki

si(mi)β^−|mi|

⎞

⎠

=

m1∈B^k1,...,mr∈B^kr

s₁(m₁)...s_r(m_r)β^{−(|m1|+...+|mr|)}

=:

∞ m=0

β^−mρ(k;m), (4.10)

where

ρ(k;m) =

m1∈Bk1,...,mr∈Bkr

|m1|+...+|mr|=m

s₁(m₁)...s_r(m_r)

is a nonnegative integer. Substituting (4.10) into (4.3), we get β^C4DP(ξ) =Q(η) =B₀+

k∈Γ

B_kη^k

=B₀+

k∈Γ

B_k ∞ m=0

β^−mρ(k;m).

LetRbe a nonnegative integer. Then β^R+C4DP(ξ) =B0β^R+

k∈Γ

Bk

∞ m=−R

β^−mρ(k;m+R).

Put

YR:=

k∈Γ

Bk

∞ m=1

β^−mρ(k;m+R),

Z_R:=B₀β^R+

k∈Γ

B_k 0 m=−R

β^−mρ(k;m+R).

Then we have

β^R+C4DP(ξ) =Y_R+Z_R. (4.11)

Note thatZR∈Z[β] because Bk∈Z[β] for any k∈{0}∪Γ.

We now introduce a sketch of the proof of Theorem3.2. We first show for any nonnegative integerR that

Z_R= 0 or |Z_R| ≥C₅β^−R1−2C

in Lemma4.2. Next, we show that there exists a positive integerR satisfying 0< YR< C5β^−R1−2C

(4.12)

in Lemma4.6. Therefore, (4.11) implies that P(ξ)= 0.

SinceP(X) is any non-constant polynomial whose coefficients are elements inZ[β], we deduce thatξ1, ..., ξr are algebraically independent.

We now give upper bounds forρ(k;m).

Lemma 4.1. For anyk∈Γ, we have log⁺ρ(k;m) =o

m^1−2C .

In particular, there exists a positive constant C6 satisfying the following: for any k∈Γandm∈N,

ρ(k;m)≤C6β^m1−2C.

Proof. Letk∈Γ. Using (4.9) and the definition ofρ(k;m), we get ρ(k;m)≤(m+1)^k1+...+kr

1≤i≤rmax

0≤n≤m

s_i(n)

k1+...+kr

≤(m+1)^D

1max≤i≤r 0≤n≤m

s_i(n) D

.

Taking the logarithm of the inequality above, we obtain, by (4.7), log⁺ρ(k;m) =o

m^1−2C

as m tends to infinity. Moreover, the second assertion also holds because Γ is a finite set.

Lemma 4.2. There exists a positive constant C₅ satisfying the following: For any nonnegative integerR, we have

ZR= 0 or |ZR| ≥C5β^−R1−2C.

Proof. Put d:=degβ. Let π1, ..., πd be the conjugate embeddings of Q(β) intoC, whereπ₁(γ)=γ for anyγ∈Q(β). Set

C₇:= max

|π_i(B_k)|i= 1, ..., d,k∈ {0}∪Γ . Settingπ_i(β)=:β_i fori=2, ..., d, we have

πi(ZR) =πi(B0)β_i^R+

k∈Γ

πi(Bk) R m=0

β_i^mρ(k;−m+R).

Recall that|βi|≤1 becauseβ is a Pisot or Salem number. Thus,

|π_i(Z_R)| ≤C₇+

k∈Γ

C₇(R+1) max

0≤m≤Rρ(k;m).

Using Lemma4.1, we get, fori=2, ..., d, log⁺|πi(ZR)|=o

R^1−2C

asR tends to infinity. Hence, we see for any sufficiently largeR that d

i=2

log⁺|π_i(Z_R)| ≤R^1−2Clogβ, and so

d i=2

|πi(ZR)| ≤β^R1−2C.

Assume thatZR=0. SinceZR is an algebraic integer, we obtain 1≤ |Z_R|

d i=2

|π_i(Z_R)|.

Therefore, we deduce for any sufficiently largeR that

|Z_R| ≥β^−R1−2C. In what follows, we consider lower bounds forYR.

Lemma 4.3. Let k∈Γ and let m be a nonnegative integer. Then ρ(k;m) is positive if and only ifm∈|k|B.

Proof. We observe thatρ(k;m) is positive if and only if there existm1∈B^k1, ..., m_r∈B^kr such that m=_r

i=1|m_i|. Thus, ρ(k;m) is positive if and only if there exists ann∈B^k1×...×B^kr such thatm=|n|,that is,|n|∈|k|BbecauseB^ki={0}for any 1≤i≤rwithki=0.

Since 0∈B, we get

B ⊂2B ⊂...⊂DB. (4.13)

Letk∈Γ and n=(n(1), ..., n(|k|))∈N^|k|. We divideninto rparts as follows:

n=

n(1), ..., n(k1)

! ", n(k1+1), ..., n(k1+k2) ! ", ...,

n(k₁+...+k_r−1+1), ..., n(k₁+...+k_r−1+k_r) ! "

=: (n(k,1),n(k,2), ...,n(k, r)), where

n(k, i) := (n(k1+...+ki−1+1), ..., n(k1+...+ki−1+ki)) (4.14)

fori=1, ..., r. Let

s(k;n) :=

r i=1

si

n(k, i) , (4.15)

wheres_i(n(k, i)) is defined by (4.9).

Lemma 4.4. Letm be any nonnegative integer withm∈(D−1)B. Then

k∈Γ\{g}

|Bk|ρ(k;m)≤1

2B_gρ(g;m).

(4.16)

Proof. Without loss of generality, we may assume that m∈DB. In fact, if m∈DB, then we see that both-hand sides of (4.16) are 0 by Lemma4.3and (4.13).

Using Lemma 4.3, we see for any k∈Γ2 that ρ(k;m)=0 by (4.13) and m∈

(D−1)B. Thus,

k∈Γ\{g}

|B_k|ρ(k;m) =

k∈Γ1

|B_k|ρ(k;m).

(4.17)

Hence, we may assume for the proof of Lemma 4.4 that Γ₁ is not empty. In par- ticular, we haver≥2. Note for anyk∈{g}∪Γ1 thatρ(k;m)>0 by Lemma4.3and

|k|=D.

Put

Ξ :=

n= (n(1), ..., n(D))∈ B^D |n|=m (=∅).

We apply notation (4.14) and (4.15) to k∈{g}∪Γ1 and n∈Ξ⊂N^|k|. Then we see for anyk∈{g}∪Γ₁that

ρ(k;m) =

n∈B^D,|n|=m

s₁

n(k,1) ...s_r

n(k, r)

=

n∈Ξ

s(k;n).

(4.18)

We first show for any fixedk=(k₁, ..., k_r)∈Γ1that ρ(k;m)<D!

C₃ρ(g;m).

(4.19) Put

l:= max{i≥1|ki=gi}.

Since|k|=|g|=D, we havel≥2 because there exists an integerawitha<lsatisfying ga<ka. Moreover, setting

τ:=D−(k_l+k_l+1+...+k_r) =k₁+k₂+...+k_l−1, we see thatτ >0.

Letn=(n(1), ..., n(D))∈Ξ. Take an integerbwith 1≤b≤τ satisfying s_l(n(b)) = max

1≤j≤τs_l(n(j)).

(4.20)

There exists a permutationσof the set {1, ..., τ}such that σ(τ) =b.

Let

(p(1), ..., p(τ)) := (n(σ(1)), ..., n(σ(τ))) and

p:= (p(1), p(2), ..., p(τ), n(τ+1), n(τ+2), ..., n(D)).

(4.21)

It is clear that p∈Ξ because p is obtained by a permutation of the components ofn. Note thatpis determined byk∈Γ₁ andn∈Ξ. Recall that

s(k;n) = r i=1

si

n(k, i)

and s(g;p) = r i=1

si

p(g, i) ,

where n(k, i)∈B^ki and p(g, i)∈B^gi for i=1, ..., r. By the definition of p, the last (D−τ)-th components ofnand pcoincide. Observing thatg_i=k_i for any i≥l+1, we see

n(k, i) =p(g, i) (4.22)

for eachi≥l+1 becausek_l+1+...+k_r≤D−rby the definition ofτ. Moreover, since g_l>k_l, we see thatn(k, l) andp(g, l) are denoted as

n(k, l) = (n(τ+1), ..., n(τ+k_l)),

p(g, l) = (p(τ−h+1), ..., p(τ), n(τ+1), ..., n(τ+kl)), respectively, whereh=gl−kl>0.Thus, we have

s_l p(g, l)

=s_l

n(k, l)^h

j=1

s_l

p(τ+1−j) . (4.23)

Hence, combining (4.22) and (4.23), we obtain s(g;p) =

r i=1

si

p(g, i)

≥sl

p(g, l) ^r

i=l+1

si

p(g, i)

=s_l

n(k, l)^h

j=1

s_l

p(τ+1−j) ^r

i=l+1

s_i n(k, i)

= h j=1

sl

p(τ+1−j)^r

i=l

si

n(k, i)

≥sl

p(τ)^r

i=l

si

n(k, i) . (4.24)

Note thatn(j)>0 for any 1≤j≤D. In fact, ifn(j₀)=0 for some 1≤j0≤D, then m=|n|=

1≤j≤D j=j0

n(j)∈(D−1)B,

which contradicts the assumption of Lemma4.4.

We now let i, j be integers with 1≤i≤l−1 and 1≤j≤τ. Applying (4.8) with u=l,v=i, andn=n(j)∈B\{0}, we see

C₃s_i(n(j))< s_l(n(j))^1/D≤s_l(n(b))^1/D

=s_l(p(τ))^1/D

by (4.20) andp(τ)=n(σ(τ))=n(b). In particular, we get C₃^τ max

1≤j≤τ 1≤i≤l−1

si(n(j))^τ< sl(p(τ))^{τ /D}≤sl(p(τ))

by 1≤τ≤D, and so C3

l−1

i=1

si(n(k, i))≤C3 l−1

i=1

1≤j≤τmax si(n(j))^ki

≤C₃^τ max

1≤j≤τ 1≤i≤l−1

s_i(n(j))^τ< s_l(p(τ)) (4.25)

byC3≥1. Combining (4.24) and (4.25), we obtain s(g;p)> C₃

l−1

i=1

s_i

n(k, i)^r

i=l

s_i n(k, i)

=C₃s(k;n).

(4.26)

Let n,n∈Ξ. We write n∼n if n is translated to n by a permutation of the components. Note thatp∼nfor anyn∈Ξ, wherepis defined by (4.21). We write p=:p(n) in order to emphasize thatpdepends onn. We denote the set of equiv- alence classes of Ξ with respect to∼by Ξ/∼. Using (4.18) and (4.26), we obtain

ρ(k;m) =

n∈Ξ

s(k;n) =

α∈Ξ/∼

n∈α

s(k;n)

< 1 C₃

α∈Ξ/∼

n∈α

s

g;p(n) . Observe for anyα∈Ξ/∼that

Cardα≤D!.

Moreover, ifn∈α, thenp(n)∈α. Hence, ρ(k;m)< 1

C3

α∈Ξ/∼

D! max

n∈αs(g;n)

≤D!

C3

α∈Ξ/∼

n∈α

s(g;n) =D!

C3

n∈Ξ

s(g;n) =D!

C3

ρ(g;m) by (4.18), which implies (4.19).

Recall that Γ₁=Λ₁ and A_g>0. Applying (4.17), (4.4), (4.19), the definition ofC₃, we deduce that

k∈Γ\{g}

|B_k|ρ(k;m) =

k∈Λ1

β^C4D|A_k|ρ(k;m)<

k∈Λ1

β^C4D|A_k|D!

C₃ρ(g;m)

=β^C4Dρ(g;m)·D!

C3

k∈Λ1

|A_k|

≤β^C4Dρ(g;m)·1 2Ag=1

2Bgρ(g;m), which implies Lemma4.4.