Jean-Louis Verger-Gaugry
email: jlverger@ujf-grenoble.fr

On gaps in Rényi β-expansions of unity for β > 1 an algebraic number

Keywords: 2000 Mathematics Subject Classification: 11B05, 11Jxx, 11J68, 11R06, 52C23 Beta-integer, beta-numeration, PV number, Salem number, Perron number, Mahler measure, Diophantine approximation, Mahler's series, mathematical quasicrystal

Let β > 1 be an algebraic number. We study the strings of zeros ("gaps") in the Rényi β-expansion d β (1) of unity which controls the set Z β of β-integers. Using a version of Liouville's inequality which extends Mahler's and Güting's approximation theorems, the strings of zeros in d β (1) are shown to exhibit a "gappiness" asymptotically bounded above by log(M(β))/ log(β), where M(β) is the Mahler measure of β. The proof of this result provides in a natural way a new classification of algebraic numbers > 1 with classes called Q (j) i which we compare to Bertrand-Mathis's classification with classes C1 to C5 (reported in an article by Blanchard). This new classification relies on the maximal asymptotic "quotient of the gap" value of the "gappy" power series associated with d β (1). As a corollary, all Salem numbers are in the class C1∪ Q

; this result is also directly proved using a recent generalization of the Thue-Siegel-Roth Theorem given by Corvaja.

Introduction

The exploration of the links between symbolic dynamics and number theory of β-expansions, when β > 1 is an algebraic number or more generally a real number, started with Bertrand-Mathis [Be1] [Be2]. Bertrand-Mathis, in Blanchard [Bl], reported a classification of real numbers according to their β-shift, using the properties of the Rényi β-expansion d β (1) of 1. A lot of questions remain open concerning the distribution of the algebraic numbers β > 1 in this classification. The Rényi β-expansion of 1 is important since it controls the β-shift [Pa] and the discrete and locally finite set [G1]. The aim of this note is to give a new Theorem (Theorem 1.1) on the gaps (strings of 0 s) in d β (1) for algebraic numbers β > 1, and investigate how it provides (partial) answers to some questions of [Bl], in particular for Salem numbers (Corollary 1.2).

Z β ⊂ R of β- integers [B-K] [E-VG] [Ga]
Theorem 1.1 provides an upper bound on the asymptotic quotient of the gap of d β (1) and is obtained by a version of Liouville's inequality extending Mahler's and Güting's approximation theorems. The proof of Theorem 1.1 turns out to be extremely instructive in itself since it leads to a new classification of the algebraic numbers β as a function of the asymptotics of the gaps in d β (1) and "intrinsic features", namely the Mahler measure M(β), of β (the definition of M(β) is recalled in Section 3). The existence of this double parametrization, symbolic and algebraic, was guessed in [Bl] p 137. This new classification complements Bertrand-Mathis's (Blanchard [Bl] pp 137-139) and both are recalled below for comparison's sake. The question whether an algebraic number β > 1 is contained in one class or another has already been discussed by many authors [START_REF] Bertrand-Mathis | Développements en base Pisot et répartition modulo 1[END_REF] [Pa] [PF] [Sc] [Sk] and depends at least upon the distribution of the conjugates of β in the complex plane. Only the conjugates of β of modulus strictly greater than unity intervene in Theorem 1.1 via the Mahler measure of β. Corollary 1.2 is readily deduced from this remark. We deduce that Salem numbers belong to C 1 ∪ C 2 ∪ Q 0 , whereas the Pisot numbers are in C 1 ∪ C 2 [Th].

[Be2] [Be3] [Bl] [Bo] [Bo1] [Bo2] [Bo3] [D-S] [FS] [Li1] [Li2]
Another proof of Corollary 1.2 consists of controlling the gaps of d β (1) by stronger Theorems of Diophantine Geometry which allow suitable collections of places of the number field Q(β) associated with the conjugates of β and the properties of d β (1) to be taken into account simultaneously. This alternative proof of Corollary 1.2, just sketched in Section 4, is obtained using the Theorem of Thue-Siegel-Roth given by Corvaja [A] [C].

Theorem 1.1. Let β > 1 be an algebraic number and M(β) be its Mahler measure. Denote by d β (1) := 0.t 1 t 2 t 3 . . ., with t i ∈ A β := {0, 1, 2, . . . , β -1 }, the Rényi β-expansion of 1. Assume that d β (1) is infinite and gappy in the following sense: there exist two sequences {m n } n≥1 , {s n } n≥0 such that

1 = s 0 ≤ m 1 < s 1 ≤ m 2 < s 2 ≤ . . . ≤ m n < s n ≤ m n+1 < s n+1 ≤ . . . with (s n -m n ) ≥ 2, t mn = 0, t sn = 0 and t i = 0 if m n < i < s n for all n ≥ 1. Then lim sup n→+∞ s n m n ≤ log(M(β)) log(β) . (1.1) Moreover, if lim inf n→+∞ (m n+1 -m n ) = +∞, then lim sup n→+∞ s n+1 -s n m n+1 -m n ≤ log(M(β)) log(β) . (1.2)
As in Ostrowski [Os] the quotient s n /m n ≥ 1 is called the quotient of the gap, relative to the nth-gap (assuming t j = 0 for all s n ≤ j ≤ m n+1 to characterize uniquely the gaps). Note that the term "lacunary" has often other meanings in literature and is not used here to describe "gappiness". Denote by L(S β ) the language of the β-shift [Bl] [Fr1] [Fr2] [Lo]. Bertrand-Mathis's classification ( [Bl] pp 137-139) is as follows: 1) is ultimately periodic but not finite. C 3 : d β (1) contains bounded strings of 0 s, but is not ultimately periodic. C 4 : d β (1) does not contain some words of L(S β ), but contains strings of 0 s with unbounded length. C 5 : d β (1) contains all words of L(S β ).

C 1 : d β (1) is finite. C 2 : d β (
Present classes of algebraic numbers, with the notations of Theorem 1.1:

Q (1) 0 : 1 = lim n→+∞ s n m n with (m n+1 -m n ) bounded. Q (2) 0 : 1 = lim n→+∞ s n m n with (s n -m n ) bounded and lim n→+∞ (m n+1 -m n ) = +∞. Q (3) 0 : 1 = lim n→+∞ s n m n with lim sup n→+∞ (s n -m n ) = +∞. Q 1 : 1 < lim sup n→+∞ s n m n < log(M(β)) log(β) . Q 2 : lim sup n→+∞ s n m n = log(M(β)) log(β) .
What are the relative proportions of each class in the whole set Q >1 of algebraic numbers β > 1 ? Comparing C 2 , C 3 and Q has shown that the class C 3 (of real numbers β > 1) has Hausdorff dimension one. We have:

• Q >1 ∩ C 2 ⊂ Q (1) 0 , • Q >1 ∩ C 3 ⊂ Q (1) 0 ∪ Q (2) 0 , with C 3 ∩ Q (3) 0 = ∅, • Q >1 ∩ C 4 ⊂ Q (3) 0 ∪ Q 1 ∪ Q 2 .
The Pisot numbers β are contained in C 1 ∪ Q

(1) 0 since they are such that d β (1) is finite or ultimately periodic (Parry [Pa], Bertrand-Mathis [START_REF] Bertrand-Mathis | Questions diverses relatives aux systèmes codés : applications au θ-shift[END_REF]). Recall that a Perron number is an algebraic integer β > 1 such that all the conjugates β (i) of β satisfy |β (i) | < β. Conversely, as shown in Lind [START_REF] Lind | The entropies of topological Markov shifts and a related class of algebraic integers[END_REF], Denker, Grillenberger, Sigmund [D-S] and Bertrand-Mathis [START_REF] Bertrand-Mathis | Développements en base θ et répartition modulo 1 de la suite (xθ n )[END_REF], if β > 1 is such that d β (1) is ultimately periodic (finite or not), then β is a Perron number. Not all Perron numbers are attained in this way: a Perron number which possesses a real conjugate greater than 1 cannot be such that d β (1) is ultimately periodic ( [Bl] p 138). Parry numbers belong to

C 1 ∪ C 2 . Let Q 0 = Q (1) 0 ∪ Q (2) 0 ∪ Q (3) 0 . Corollary 1.2. Let β > 1 be a Salem number which does not belong to C 1 . Then β belongs to the class Q 0 . The attribution of Salem numbers to C 1 , Q (1) 0 , Q (2) 0 and Q (3)
0 is an open problem in general, except in low degree. Boyd [Bo] [Bo3] has shown that Salem numbers of degree 4 belong to C 2 , hence to Q

(1) 0 . It is also the case of some Salem numbers of degree 6 and ≥ 8 in the framework of a probabilistic model [Bo2] [Bo3]. In Section 5 we ask the question whether Corollary 1.2 could still be true for Perron numbers.

The definition of the class Q 0 does not make any allusion to β, i.e. to M(β), to the conjugates of β, to the minimal polynomial of β or to its length, etc, but takes only into account the quotients of the gaps in d β (1). Hence this class Q 0 can be applied to real numbers β > 1 in full generality instead of only to algebraic numbers > 1. The question whether there exist transcendental numbers β > 1 which belong to the class Q 0 was asked in [Bl]; what proportion appears in each subclass ? Examples of transcendental numbers (Komornik-Loreti constant [AC] [KL], Sturmian numbers [CK]) in Q 0 are given in Section 5.

In the present note, we deal with the algebraicity of values of "gapppy" series, deduced from d β (1), at the algebraic point β -1 . In a related context, more related to transcendency, Nishioka [N] and Corvaja Zannier [CZ] have followed different paths and applied the Subspace Theorem [Sw] to deduce different results.

Definitions

For x ∈ R the integer part of x is x and its fractional part {x} = xx . The smallest integer larger than or equal to x is denoted by x . For β > 1 a real number and z ∈ [0, 1] we denote by T β (z) = βz (mod 1) the β-transform on [0, 1] associated with β [Pa] [Re], and iteratively, for all integers j ≥ 0,

T j+1 β (z) := T β (T j β (z))
, where by convention T 0 β = Id. Let β > 1 be a real number. A beta-representation (or β-representation, or representation in base β) of a real number x ≥ 0 is given by an infinite sequence (x i ) i≥0 and an integer k ∈ Z such that x = +∞ i=0 x i β -i+k , where the digits x i belong to a given alphabet (⊂ N) [START_REF] Ch | Number Representation and Finite Automata[END_REF] [Fr2] [Lo]. Among all the beta-representations of a real number x ≥ 0, x = 1, there exists a particular one called Rényi β-expansion, which is obtained via the greedy algorithm: in this case, k satisfies β k ≤ x < β k+1 and the digits

x i := β T i β x β k+1 i = 0, 1, 2, . . . , (2.1)
belong to the finite canonical alphabet

A β := {0, 1, 2, . . . , β -1 }. If β is an integer, then A β := {0, 1, 2, . . . , β -1}; if β is not an integer, then A β := {0, 1, 2, . . . , β }.
We denote by

x β := x 0 x 1 x 2 . . . x k . x k+1 x k+2 . . . (2.2)
the couple formed by the string of digits x 0 x 1 x 2 . . . x k x k+1 x k+2 . . . and the position of the dot, which is at the k-th position (between x k and x k+1 ). By definition the integer part (in base β) of x is

k i=0 x i β -i+k and its fractional part (in base β) is +∞ i=k+1 x i β -i+k . If a Rényi β-
expansion ends in infinitely many zeros, it is said to be finite and the ending zeros are omitted. If it is periodic after a certain rank, it is said to be eventually periodic (the period is the smallest finite string of digits possible, assumed not to be a string of zeros); for the substitutive approach see [F] [PF].

The Rényi β-expansion which plays an important role in the theory is the Rényi β-expansion of 1, denoted by d β (1) and defined as follows: since β 0 ≤ 1 < β, the value T β (1/β) is set to 1 by convention. Then using the formulae (2.1)

t 1 = β , t 2 = β{β} , t 3 = β{β{β}} , . . . (2.3)
The writing

d β (1) = 0.t 1 t 2 t 3 . . . corresponds to 1 = +∞ i=1 t i β -i .
Links between the set Z β of beta-integers and

d β (1) are evoked in [E-VG] [F-K] [G1] [G2] [V1] [V2]. A real number β > 1 such that d β (1
) is finite or eventually periodic is called a beta-number or more recently a Parry number (this recent terminology appears in [E-VG]). In particular, it is called a simple beta-number or a simple Parry number when d β (1) is finite. Beta-numbers (Parry numbers) are algebraic integers [Pa] and all their conjugates lie within a compact subset which looks like a fractal in the complex plane [So]. The conjugates of beta-numbers are all bounded above in modulus by the golden mean

1 2 (1 + √ 5) [So] [F-P].
3 Proof of Theorem 1.1

Since algebraic numbers β > 1 for which the Rényi β-expansion d β (1) of 1 is finite are excluded, we may consider that β does not belong to N.

Indeed, if β = h ∈ N, then d h (1) = 0.h is finite (Lothaire [Lo], Chap. 7). If β ∈ N, then β -1 = β and the alphabet A β equals {0, 1, 2, . . . , β },
where β denotes the greatest integer smaller than or equal to β.

Let f (z) := +∞ i=1 t i z i be the "gappy" power series deduced from the representation d β (1) = 0.t 1 t 2 t 3 . . . associated with the β-shift (gappy in the sense of Theorem 1.1). Since d β (1) is assumed to be infinite, its radius of convergence is 1. By definition, it satisfies

f (β -1 ) = 1, (3.1)
which means that the function value f (β -1 ) is algebraic, equal to 1, at the real algebraic number β -1 in the open disk of convergence D(0, 1) of f (z) in the complex plane. This fact is a general intrinsic feature of the Rényi expansion process which leads to the following important consequence by the theory of admissible power series of Mahler [Ma].

Proposition 3.1.

lim sup n→+∞ s n m n < +∞. (3.2)
Proof. This is a consequence of Theorem 1 in [Ma]. Indeed, if we assume that there exists a sequence of integers (n i ) which tends to infinity such that lim i→+∞ s ni /m ni = +∞, then f (z) would be admissible in the sense of [Ma]. Since f (z) is a power series with nonnegative coefficients, which is not a polynomial, the function value f (β -1 ) should not be algebraic. But it equals 1. Contradiction.

Let us improve Proposition 3.1. Assume that lim sup

s n m n > log(M(β)) log(β) (3.3)
and show the contradiction with (1.1) and (1.2). Recall that, if

P β (X) = d i=0 α i X i = α d d-1 i=0 (X -β (i) )
with d ≥ 1, α 0 α d = 0, denotes the minimal polynomial of β = β (0) > 1, having β (1) , β (2) , . . . , β (d-1) as conjugates, the Mahler measure of β is by definition

M(β) := |α d | d-1 i=0 max{1, |β (i) |}.
Güting [Gü] has shown that the approximation of algebraic numbers by algebraic numbers is fairly difficult to realize by polynomials. In the present proof, we use a version of Liouville's inequality which generalizes approximation theorems obtained by Güting [Gü], and apply it to the values of the "polynomial tails" of the power series f (z) at the algebraic number β -1 , to obtain the contradiction. Let us write

f (z) = +∞ n=0 Q n (z) (3.4) with Q n (z) := mn+1 i=sn t i z i , n = 0, 1, 2, . . . . (3.5)
By construction the polynomials Q n (z), of degree m n+1 , are not identically zero and Q n (1) > 0 is an integer for all n ≥ 0. Denote by S n (z) = -1 + mn i=1 t i z i the m n th-section polynomial of the power series f (z)-1 for all n ≥ 1. Recall that, for R(

X) = v i=0 α i X i ∈ Z[X], L(R) := v i=0 |α i | denotes the length of the polynomial R(X). We have: L(S n ) = 1 + mn i=1 t i = 1 + n-1 j=0 Q j (1)
. From Theorem 5 in [Gü] we deduce that only one of the following cases (G-i) or (G-ii) holds, for all n ≥ 1:

(G -i) S n (β -1 ) = 0, (3.6) (G -ii) S n (β -1 ) ≥ 1 1 + n-1 j=0 Q j (1) d-1 L(P * β )
mn , (3.7)

where P * β (X) = X d P β (1/X) is the reciprocal polynomial of the minimal polynomial of β, for which L(P β ) = L(P * β ) ∈ N \ {0, 1}. Case (G-i) is impossible for any n. Indeed, if there exists an integer n 0 ≥ 1 such that (G-i) holds, then, since all the digits t i are positive and that β -1 > 0, we would have t i = 0 for all i ≥ s n0 . This would mean that the Rényi expansion of 1 in base β is finite, which is excluded by assumption. Contradiction. Therefore, the only possibility is (G-ii), which holds for all integers n ≥ 1. From Lemma 3.10 and Liouville's inequality (Proposition 3.14) in Waldschmidt [W] the inequality (G-ii) can be improved to

(L -ii) S n (β -1 ) ≥ 1 1 + n-1 j=0 Q j (1) d-1 (M(β)) mn . (3.8)
This improvement may be important; recall the well-known inequalities:

M(β) ≤ L(P β ) ≤ 2 deg(β) M(β)
and see [W] p113 for comparison with different heights. On the other hand, since S n (β -1 ) = +∞ i=sn t i β -i for all integers n ≥ 1, we deduce

S n (β -1 ) ≤ β 1 -β -1 β -sn n = 1, 2, . . . . (3.9)
Putting together (3.8) and (3.9), we deduce that

β sn 1 + n-1 j=0 Q j (1) d-1 M(β) mn ≤ β 1 -β -1
(3.10) should be satisfied for n = 1, 2, 3, . . .. Denote

u n := β sn 1 + n-1 j=0 Q j (1) d-1
M(β) mn for all n ≥ 1.

Proof of (1.1): from (3.3) assumed to be true there exists a sequence of integers (n i ) which tends to infinity and an integer i 0 such that

s ni m ni > log(M(β)) log(β) for all i ≥ i 0 . Now, 1 1 + β m ni d-1   β sn i mn i M(β)   mn i ≤ 1 1 + ni-1 j=0 Q j (1) d-1   β sn i mn i M(β)   mn i ≤ u ni . (3.11) For i ≥ i 0 the inequality 1 = β log(M(β)) log(β) M(β) < β sn i mn i M(β) (3.12)
holds. This implies that the left-hand side member of (3.11) tends exponentially to infinity when i tends to infinity. By (3.11) this forces u ni to tend to infinity. The contradiction now comes from (3.10) since the sequence (u n ) should be uniformly bounded.

Proof of (1.2): for n = 1, 2, . . ., let us rewrite the n-th quotient

u n+1 u n = β sn+1-sn M(β) mn+1-mn 1 + n-1 j=0 Q j (1) d-1 1 + n j=0 Q j (1) d-1 (3.13) as u n+1 u n = β s n+1 -sn m n+1 -mn M(β) mn+1-mn (m n+1 -m n + 1) (d-1)   (mn+1 -m n + 1) (d-1) 1 + n-1 j=0 Q j (1) d-1 1 + n j=0 Q j (1) d-1   
(3.14) and denote

U n := 1 (m n+1 -m n + 1) (d-1)   β s n+1 -sn m n+1 -mn M(β)   mn+1-mn (3.15) and W n := (m n+1 -m n + 1) (d-1) 1 + n-1 j=0 Q j (1) 1 + n j=0 Q j (1) d-1 (3.16) so that u n+1 /u n = U n W n . Lemma 3.2. 0 < lim inf n→+∞ W n
(3.17)

Proof. Assume the contrary. Then there exists a subsequence (n i ) of integers which tends to infinity such that lim i→+∞ W ni = 0. In other terms, for all > 0, there exists i

1 such that i ≥ i 1 implies W ni ≤ , equivalently (m ni+1 -m ni + 1) 1 + ni-1 j=0 Q j (1) ≤ 1 d-1 × 1 + ni j=0 Q j (1) . (3.18)
Since, by hypothesis, t sn ≥ 1 and t mn+1 ≥ 1 for all n ≥ 1, we have:

n i ≤ 1 + ni-1 j=0 Q j (1). On the other hand, Q ni (1) ≤ β (m ni+1 -m ni + 1)
. Then, from (3.18) with taken equal to 1, we would have

n i ≤ 1+ ni-1 j=0 Q j (1) ≤ Q ni (1) (m ni+1 -m ni + 1) -1 ≤ β × m ni+1 -m ni + 1 m ni+1 -m ni ≤ 3 2 β .
(3.19) But the left-hand side member of (3.19) tends to infinity which is impossible. Contradiction.

Let us assume that (1.2) does not hold and show the contradiction ; that is, assume that lim inf n→+∞ (m n+1m n ) = +∞ and lim sup n→+∞ (s n+1 -

s n )/(m n+1 -m n ) > log(M(β))/ log(β) hold. Then 1 = β log(M(β)) log(β) M(β) < β s n i +1 -sn i m n i +1 -mn i M(β) (3.20)
for some sequence of integers (n i ) which tends to infinity. This proves that lim sup n→+∞ U n = +∞ since lim i→+∞ U ni = +∞ exponentially, by (3.15) and (3.20). By Lemma 3.2 there exists r > 0 such that W n ≥ r for all n large enough. Therefore, u n+1 /u n = U n W n ≥ rU n for all n large enough. Since lim sup n→+∞ U n = +∞ we conclude that lim sup u n+1 /u n = +∞, hence that lim sup u n = +∞. This contradicts (3.10) and proves (1.2). leads to a contradiction. Denote by K the algebraic number field Q(β), considered as a multivalued field with the product formula [C] [Sw] (see also [Lg]).

The present proof is merely an adaptation of that of Theorem 1 in [A], though the aims are different, and therefore does not merit publication. We simply point out a few hints for the interested reader.

The main result which is used is Corollary 1 of the Main Theorem in [C], as in [A]. This is a version of the Thue-Siegel-Roth Theorem given by Corvaja which is stronger than Roth Theorem for number fields [Le] [Sw] to the extent it allows us to introduce a missing proportion of places of K by considering the projective approximation of the point at infinity in P 1 (K). Since β is a Salem number, it is a unit [B-S]. Hence, this missing proportion has just to be chosen among the pairwise distinct Archimedean places of K.

5 On the class Q 0

Perron numbers

Let us give, after Solomyak ([So], p 483), the example of a Perron number which is not a beta-number and therefore which is not in the class C 2 , without knowing whether it is in the class Q 0 . This example allows us to estimate the sharpness of the upper bound log(M(β))/ log(β) in (1.1). Recall that a real number β > 1 is a beta-number if the orbit of x = 1 under the transformation T β : x → βx (mod 1) is finite [Lo] [PF]. The set of all conjugates of all betanumbers is the union of the closed unit disc in the complex plane and the set of reciprocals of zeros of the function class {f (z) = 1 + a j z j | 0 ≤ a j ≤ 1}. The closure of this domain, say Φ, is compact and was studied by Flatto, Lagarias and Poonen [F-P] and Solomyak [So]. After [So], the Perron number

β = 1 2 (1 + √ 13), dominant root of P β (X) = X 2 -X -3, is not a beta-number, though its only conjugate β = 1 2 (1 - √ 13
) lies in the interior int(Φ). We have M(β) = 3. By Theorem 1.1 the "quotients of the gaps" are asymptotically bounded above by log(3)/ log(β) = 1.3171 . . ., a much better bound than the degree d = 2 of β (see Lemma 5.1). This does not suffice to conclude that 1 2 (1 + √ 13) belongs to Q 0 . Do all Perron numbers belong to Q 0 ? Let β > 1 be a Perron number of degree d ≥ 2 and denote by β (1) , β (2) , . . . , β (d-1) the conjugates of β = β (0) , roots of the minimal polynomial P β (X) of β. Let K β := max{|β (i) | | i = 1, 2, . . . , d -1}.

Lemma 5.1. Let n = n β (with 2 ≤ n β ≤ d) be the number of conjugates of β of modulus strictly greater than unity (including β). Then

log(M(β)) log(β) ≤ n - n -1 (dβ) 6d 3 log β . (5.1) Proof. Obvious since (Lemma 2 in [Li2]): K β < β 1 - 1 (dβ) 6d 3 .
The upper bound (5.1) does not allow us to give a positive answer to the question and has probably to be improved.

Transcendental numbers

Let us show that the Komornik-Loreti constant [KL] [AC] belongs to Q

(1) 0 .

Theorem 5.2. There exists a smallest q ∈ (1, 2) for which there exists a unique expansion of 1 as 1 = ∞ n=1 δ n q -n , with δ n ∈ {0, 1}. Furthermore, for this smallest q, the coefficient δ n is equal to 0 (respectively, 1) if the sum of the binary digits of n is even (respectively, odd). This number q can then be obtained as the unique positive solution of 1 = ∞ n=1 δ n q -n . It is equal to 1.787231650 . . .. This constant q is named Komornik-Loreti constant. Allouche and Cosnard [AC] have shown the following result.

Theorem 5.3. The constant q is a transcendental number, where the sequence of coefficients (δ n ) n≥1 is the Prouhet-Thue-Morse sequence on the alphabet {0, 1}.

The uniqueness of the development of 1 in base q given by Theorem 5.2 allows us to write d q (1) = 0.δ 1 δ 2 δ 3 . . . , the coefficients δ n being the digits of the Rényi q-expansion of 1. Since the strings of zeros and 1's in the Prouhet-Thue-Morse sequence are known (Thue, 1906(Thue, /1912;;[AS]) and uniformly bounded, the constant q belongs to the class Q

(1) 0 .

As second example, let us show that Sturmian numbers in the interval (1, 2) (in the sense of [CK]) belong to Q Let us consider all the Sturmian numbers β ∈ (1, 2) for which the twoletter alphabet is {0, 1}. For such numbers gappiness appears in d β (1) (in the sense of Theorem 1.1). By Theorem 3.3 in [CK] strings of zeros, resp. of 1's, cannot be arbitrarily long. This gives the claim. discussions and remarks. The author would like to thank Catriona MacLean for her careful rereading of the manuscript.

  those β which give ultimate periodicity in d β (1) and those for which d β (1) is not ultimately periodic ? Schmeling ([Sc] Theorem A)

4

  A direct proof of Corollary 1.2 Let β > 1 be a Salem number such that β ∈ C 1 . Using the notations of Theorem 1.1 we show that the

  β > 1 is called a Sturmian number if d β (1) is a Sturmian word over a binary alphabet {a, b}, with 0 ≤ a < b = β . Chi and Kwon [CK] have shown the following theorem. Theorem 5.4. Every Sturmian number is transcendental.

Acknowledgments

The author thanks J.-P. Allouche, B. Adamczewski, J. Bernat, V. Berthé, C. Frougny, J.-P. Gazeau and the anonymous referee for valuable comments,


Gabriella A Figueiredo

Rafael M P Paulo

Ana M F Barela

Cédrick Bonnet

Dr José A Barela
email: jose.barela@unesp.br

Trunk and head displacements stabilized to perform both horizontal and vertical saccadic eye movements

Keywords: Postural control, Vision, Saccades, Coordination

Vision is crucial for humans to interact with their surrounding environment, and postural sway is reduced to allow short saccadic eye movements. However, the extent of subtle changes in postural control for horizontal and vertical saccadic eye movements remains unclear. The goal of this study was to investigate the effects of vertical and horizontal saccadic eye movements on head and trunk control in young adults. Fifteen healthy adults (23.4±4.7 years) stood upright in three conditions for 60 s: fixation, horizontal, and vertical guided eye movements. In fixation, participants had to fixate on a stationary target. In both the horizontal and vertical eye movements, the target was presented with a frequency of 0.5 Hz and a visual angle of 11 °. Eye displacement was monitored using a SMI eye tracker (ETG2.0) and trunk and head sway were monitored using infrared markers (Optotrak 3020, NDI). The mean sway amplitude was lower in both directions for eye movements and lowest in the vertical direction compared to the fixation condition. The sway area was also lower in vertical movement than in the fixation condition. We also found that the reduction was greater at head level than at trunk level. The median frequency sway in the anterior-posterior direction was higher in both eye movements than in fixation. Based upon these results, we suggest that to perform short saccadic eye movements, postural sway is more strongly controlled at the head level than at the trunk and in vertical saccadic eye movements than in horizontal movements.

Introduction

Vision is continuously used to explore and obtain information to guide body movement in the environment. Visual information also plays an important role in postural control and the maintenance of balance [START_REF] Lee | Visual proprioceptive control of stance[END_REF]. In the 20 th century, body sway magnitude was shown to be significantly reduced in the fixation of a stationary target with open eyes (fixation tasks) compared to when the eyes were closed [START_REF] Paulus | Visual stabilization of posture: physiological stimulus characteristics and clinical aspects[END_REF]. More recently, in the 21 st century, researchers have consistently shown that body sway is reduced to perform visually guided saccadic eye movements in the upright stance, in comparison to fixation in young adults [START_REF] Stoffregen | Postural stabilization of visually guided eye movements[END_REF][START_REF] Rey | Saccades horizontal or vertical at near or at far do not deteriorate postural control[END_REF][START_REF] Bonnet | Active vision task and postural control in healthy, young adults: Synergy and probably not duality[END_REF][START_REF] Thomas | Eye Movements Affect Postural Control in Young and Older Females[END_REF], elderly people [START_REF] Aguiar | Effects of saccadic eye movements on postural control in older adults[END_REF], and children [START_REF] Ajrezo | Saccades improve postural control: a developmental study in normal children[END_REF][START_REF] Bucci | Effect of visual attention on postural control in children with attention-deficit/hyperactivity disorder[END_REF].

It is now well accepted that a reduction in postural sway in guided eye movements leads to increased visual stability to perform more spatially accurate saccades [START_REF] Rodrigues | Effects of saccadic eye movements on postural control stabilization[END_REF][START_REF] Bonnet | Active vision task and postural control in healthy, young adults: Synergy and probably not duality[END_REF]. When guided eye movements are performed in the upright standing position, postural sway is reduced to allow accurate gaze shifts, which is an indication of a functional relationship between posture and gaze control [START_REF] Stoffregen | Postural stabilization of visually guided eye movements[END_REF]). This functional relationship has been suggested to be attained by afferent (minimizing the changes in the projected image on the retina) and efferent (attenuating postural sway in an attempt to connect pre-and postsaccadic views of the scene) mechanisms [START_REF] Guerraz | Ocular versus extraocular control of posture and equilibrium[END_REF]. Another possibility is that improvement of postural control to perform eye movements is due to increased proprioceptive signals from extra-ocular muscles and/or a shift in attention [START_REF] Rey | Saccades horizontal or vertical at near or at far do not deteriorate postural control[END_REF].

Many researchers have examined postural control in horizontal eye movement tasks versus control fixation tasks [START_REF] Stoffregen | Postural stabilization of visually guided eye movements[END_REF][START_REF] Stoffregen | Postural sway and the frequency of horizontal eye movements[END_REF][START_REF] Rey | Saccades horizontal or vertical at near or at far do not deteriorate postural control[END_REF][START_REF] Fukushima | Postural control during downward sway head movements in young subjects[END_REF][START_REF] Irving | Difference between vertical and horizontal saccades across the human lifespan[END_REF][START_REF] Polastri | Semi tandem base of support degrades both saccadic gaze control and postural stability particularly in older adults[END_REF]). Much less is known about eye movements in the vertical direction, which may be due to the greater importance attributed to horizontal eye movements because we use them more often in ongoing activities [START_REF] Foulsham | Saccade control in natural images is shaped by the information visible at fixation: evidence from asymmetric gaze-contingent windows[END_REF], for example, when we read text. In an earlier publication, authors suggested that the mechanisms for horizontal and vertical saccades should be similar [START_REF] Smit | A parametric analysis of human saccades in different experimental paradigms[END_REF]. However, we presume that the results of postural control are different in horizontal and vertical guided eye movements. Indeed, the two orientations of motion are both neurophysiological and anatomically distinct [START_REF] Leigh | A survey of eye movements: characteristics and teleology[END_REF]. Moreover, performing vertical eye movements might increase head rotation more than during horizontal eye movements, as shown in previous studies [START_REF] Fukushima | Postural control during downward sway head movements in young subjects[END_REF][START_REF] Polastri | Semi tandem base of support degrades both saccadic gaze control and postural stability particularly in older adults[END_REF].

The lack of or little attention to the use of vertical eye movements for postural control is somewhat surprising since, in our environment, the vertical motion of objects is common and in line with gravity. For instance, gravity-congruent visual motion is associated with reduced sway compared to gravity-incongruent and horizontal stimulus motion in adults standing upright [START_REF] Balestrucci | Effects of visual motion consistent or inconsistent with gravity on postural sway[END_REF]. Based upon these findings, it was suggested that dynamic visual cues coherent with gravitational field direction are functionally used for postural control and that the neural processing of the visual cues associated with other sensory cues (such as vestibular cues) provides a predictive and matching internal model that improves postural control [START_REF] Balestrucci | Effects of visual motion consistent or inconsistent with gravity on postural sway[END_REF]. In contrast, postural stability did not improve when young adults were asked to follow moving dots in the horizontal and vertical directions when standing on an unstable platform [START_REF] Vagaja | The influence of saccadic eye movements on postural stability during standing on an unstable platform[END_REF].

In brief, new information needs to be uncovered regarding the use of vertical eye movements in postural control, especially in comparison with horizontal eye movements. Therefore, the aim of this study was to investigate the effects of guided eye movements in vertical and horizontal directions on head and trunk control in young adults. We hypothesized that the attenuation effect on the magnitude of sway would occur for both horizontal and vertical eye movement conditions but would be larger in horizontal eye movement conditions, since horizontal eye movements are more often employed in the daily activities [START_REF] Foulsham | Saccade control in natural images is shaped by the information visible at fixation: evidence from asymmetric gaze-contingent windows[END_REF]. Furthermore, we expected, assuming an inverted pendulum model in maintaining an upright stance [START_REF] Winter | Stiffness Control of Balance in Quiet Standing[END_REF], that head sway would be larger than trunk sway and that postural sway would be reduced as a whole and not only at the head level.

Methods

Participants

Fifteen healthy undergraduate students (23.4 ± 4.7 years, 6 males and 9 females) participated in this study. All participants had normal vision or corrected-to-normal vision with glasses or contact lenses. Participants had no knowledge of the purpose of the experiment and reported no history of falls, dizziness, or postural instability. Prior to the experimental procedures, the participants signed a written consent form with all procedures approved by the Institutional Ethics Committee.

Procedures

Participants wore an eye movement tracking system (Eye Tracking Glasses -ETG2.0 SMI) to capture eye displacement (iViewETG SMIversion 27.1). Four IRED markers (Optotrak 3020 -NDI) were attached [START_REF] Rodrigues | Effects of saccadic eye movements on postural control stabilization[END_REF][START_REF] Rodrigues | Saccadic and smooth pursuit eye movements attenuate postural sway similarly[END_REF] to the participant: one on the participant's back (at approximately the 8 th thoracic vertebra) and three in a triangle shape cluster (Smart Marker Rigid Body -NDI) that was strapped to the participant's head. Tridimensional coordinates from the trunk and from one marker on the head (the one on the top of the triangle shape cluster) were used to analyze trunk and head sway in the anterior-posterior (AP) and mediallateral (ML) directions.

Participants stood inside a small room (0.90 x 1.10 x 1.85 m in width, depth, and height, respectively) with black walls. The eye-movement tracking system was then calibrated when participants fixated specific target and eye positioning was indicated and recorded. During recordings, participants were instructed to stand as still as possible, with their feet parallel and spaced apart at pelvis width, and their arms hanging at their sides. Participants performed three experimental conditions: (1) fixation, (2) horizontal, and (3) vertical guided eye movements. In the three conditions, participants were 1 m away from the monitor (LG, model Flatron L1753T8) positioned at their eye level. In the fixation condition, participants fixated on a target (a 1.5-cm diameter black circle, with a visual angle of approximately 1.15 °) displayed in the center of the monitor and surrounded by a white background. In the horizontal and vertical guided eye movement conditions, participants gazed at a target that appeared and disappeared on the left and right, as well as the top and bottom sides of the monitor (9.75 cm from the monitor center), at a visual angle of 11 ° in the horizontal and vertical directions, at a frequency of 0.5 Hz, controlled by specific software (Flash Mx, version 6.0) [START_REF] Rodrigues | Effects of saccadic eye movements on postural control stabilization[END_REF][START_REF] Rodrigues | Saccadic and smooth pursuit eye movements attenuate postural sway similarly[END_REF]. In both cases, only eye movements (and no head displacement) were necessary to perform the task because of the short distance between the participants and the target, and the small visual angle of the appearance of the stimulus [START_REF] Rodrigues | Effects of saccadic eye movements on postural control stabilization[END_REF]. It is well known that only eye movements are useful to perform eye movements lower than 15° .

Each condition was repeated three times, with each participant performing a total of nine trials. The trials were randomized in blocks of three (one block for each condition). Eye displacement was also monitored online, throughout each trial inspecting eye position made available by the eye tracking system, to ensure compliance with the instructions on how to perform each experimental condition. Based on this online inspection, one of the experimental investigators was able to visually confirm that all participants performed the tasks as required.

Data analysis

For postural sway, tridimensional coordinates for the trunk and head markers were filtered using a second-order digital Butterworth filter with a cutoff frequency of 5 Hz. Using the displacement of these two markers in the ML and AP directions, the following sway variables were calculated: sway area, mean sway amplitude, and median sway frequency. The sway area corresponded to the 95% confidence ellipse area of the head and trunk dispersion when the displacements of both directions were considered together. The mean sway amplitude, which was used to estimate trunk and head stability in each direction, was obtained by removing a linear trend from each data point of the individual ML and AP time series and then calculating the standard deviation. High or low values of the mean sway amplitude indicated less or more trunk and head stability, respectively. The median sway frequency was obtained by performing a fast Fourier transformation of the ML and AP time-series data and obtaining the corresponding frequency to half of the total power spectrum area for each trial. The mean sway frequency was used to further examine postural control. In this regard, increasing or decreasing values indicated better or worse postural control, respectively, as a strategy to reduce or increase postural sway.

Finally, Pearson's linear correlation coefficients were obtained between trunk and head displacements, for both AP and ML directions. Coefficient values close to 1 indicate that trunk and head were dependent to each other, meaning that both segments sway together.

Statistical analysis

After identifying the normality and homogeneity of variance assumptions, an analysis of variance (ANOVA) and two multivariate analyses of variance (MANOVA) were employed with visual condition (fixation, and horizontal and vertical eye guided movements) and body segment (trunk and head) as factors; both were treated as repeated measures. The dependent variable for the ANOVA was the sway area, and the dependent variables for the two MANOVAs were the mean sway amplitude and the median frequency sway for both ML and AP directions. Finally, another MANOVA was employed with visual condition as factor, also treated as repeated measure, and dependent variables were the correlation coefficients between trunk and head displacements for both AP and ML directions. When necessary, univariate analyses and post hoc tests with Bonferroni adjustments were performed. All analyses were performed using SPSS software, and ANOVA/MANOVAs were performed with the significance level set at 0.05.

Results

The participants were able to perform fixation and guided eye movements to the specific target well. Fig 1 depicts a representative time series of gaze position for the horizontal and vertical directions in all three conditions, illustrating task accomplishment by the participant.

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Postural control

Fig 2 depicts the sway area for the trunk and head for all three conditions. ANOVA showed a significant difference for condition (F2,28 = 7.54, p = 0.002, ηp 2 = 0.350) and body segment (F1,14 = 21.80, p < 0.001, ηp 2 = 0.609), but there was no significant interaction between the two. Post hoc tests showed that the sway area, for both the trunk and head, was significantly lower in vertical guided eye movement than in the fixation condition. Moreover, the sway area was significantly larger for the head than for the trunk.

Please insert Figure 2 here

Fig 3 depicts the median sway amplitude for both the trunk and head in all three visual conditions. MANOVA revealed a significant difference for condition (Wilks' Lambda = 0.496, F4,54 = 5.67, p = 0.001, ηp 2 = 0.296) and body segment (Wilks' Lambda = 0.169, F2,13 = 31.92, p < 0.001; ηp 2 = 0.831), but there was no significant interaction. Univariate analyses showed a significant condition effect for both the AP (F2,28 = 9.36, p = 0.001, ηp 2 = 0.401) and ML (F2,28 = 3.48, p = 0.04, ηp 2 = 0.199) directions. In the AP direction, post hoc tests indicated that the mean sway amplitude was significantly larger in the fixation condition than in the horizontal and vertical guided eye movements. No difference was observed between the horizontal and vertical guided eye movement conditions (p > 0.05). For the ML direction, post hoc tests indicated that the mean sway amplitude was significantly larger in the horizontal than in the vertical guided eye movement. Univariate analyses showed body segment effects for both AP (F1,14 = 60.62, p < 0.001, ηp 2 = 0.812) and ML (F1,14 = 17.72, p = 0.001, ηp 2 = 0.559). The mean sway amplitude was significantly larger for the head than the trunk segment for both the AP and ML sway directions.

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Fig 4 depicts the median sway frequency for both the trunk and head in all three visual conditions. MANOVA revealed a significant difference for condition (Wilks' Lambda = 0.413, F4,11 = 3.90, p = 0,033, ηp 2 = 0,587) and body segment (Wilks' Lambda = 0.600, F2,13 = 4.34, p = 0.036, ηp 2 = 0.400) but no significant condition or body segment interaction. Univariate analyses showed a significant condition effect only in the AP direction (F2,28 = 7.04, p = 0.003, ηp 2 = 0.335). Post hoc tests indicated that the median sway frequency was significantly lower in the fixation condition than in the horizontal and vertical guided eye movement conditions. Univariate analyses showed no significant body segment effects in either the AP or ML directions (p > 0.05).

Please insert Figure 4 here

Table 1 depicts the correlation coefficients between the trunk and the head in all three visual conditions for both AP and ML directions. MANOVA did not reveal a significant difference for condition (Wilks' Lambda = 0.662, F4,11 = 1.40, p > 0.05, ηp 2 = 0.338). Coefficient values (≥ 0.90) indicated that the trunk and the head sway were related to each other and the lack of difference among conditions indicated that both trunk and head segments synchronization was similar in all the eye movement conditions.

Please insert Table 1 here

Discussion

The purpose of this study was to examine the control of trunk and head movements to perform saccadic eye movements in horizontal and vertical directions in young adults. Our results showed that postural sway was reduced in both vertical and horizontal eye movements, but more so in the vertical movements. The median sway frequency increased in the AP axis in both eye movement conditions. Consistent with our second hypothesis, head sway was greater than trunk sway.

Sway magnitude reduction to perform guided eye movements has been observed in many studies involving horizontal eye movements in young adults [START_REF] Stoffregen | Postural stabilization of visually guided eye movements[END_REF][START_REF] Rey | Saccades horizontal or vertical at near or at far do not deteriorate postural control[END_REF][START_REF] Rodrigues | Effects of saccadic eye movements on postural control stabilization[END_REF][START_REF] Thomas | Eye Movements Affect Postural Control in Young and Older Females[END_REF], elderly people [START_REF] Aguiar | Effects of saccadic eye movements on postural control in older adults[END_REF], and children [START_REF] Ajrezo | Saccades improve postural control: a developmental study in normal children[END_REF][START_REF] Bucci | Effect of visual attention on postural control in children with attention-deficit/hyperactivity disorder[END_REF]. Several researchers have also shown a reduction in sway magnitude when vertical eye movements are performed [START_REF] Rodrigues | Saccadic and smooth pursuit eye movements attenuate postural sway similarly[END_REF][START_REF] Polastri | Semi tandem base of support degrades both saccadic gaze control and postural stability particularly in older adults[END_REF]. Based upon our results, we consistently showed a significant reduction in the mean sway amplitude in the AP direction (Fig 3) in both horizontal and vertical eye movements and, therefore, we validated our hypothesis that stated reduction of sway magnitude for both horizontal and vertical eye movements.

Originally, we expected that the values for body sway stabilization would be larger to preferentially perform horizontal saccadic eye movements. Our results, however, led us to invalidate our hypothesis because they showed that postural stabilization was more pronounced in performing vertical eye movements (Fig 2 and3). In fact, our results showed that the reduction in sway area was only observed in the vertical eye movement condition, not in the horizontal eye movement condition, compared to the fixation condition (Fig 2). Moreover, in the vertical eye movement condition vs. horizontal eye movements, we observed a greater reduction in mean sway amplitude in the ML direction (Fig 3). Based upon these results, body sway stabilization may be more useful in performing vertical guided eye movements than horizontal ones. Hence, we do not corroborate previous results obtained when young adults stood on an unstable surface, or when there was no difference in stability between vertical and horizontal moving visual stimuli [START_REF] Vagaja | The influence of saccadic eye movements on postural stability during standing on an unstable platform[END_REF]. On the other hand, visual motions congruent with the gravity line provided further reduction of postural sway magnitude in adults standing upright and suggested that visual dynamic gravitational cues seem to be involved in postural control functioning [START_REF] Balestrucci | Effects of visual motion consistent or inconsistent with gravity on postural sway[END_REF].

Postural control was improved to successfully perform vertical eye movements despite the predominance of horizontal eye movements in daily life activities [START_REF] Foulsham | Saccade control in natural images is shaped by the information visible at fixation: evidence from asymmetric gaze-contingent windows[END_REF]. One reason for this may be that postural control is more unstable, that is, postural sway may increase more easily in performing up-down rather than left-right saccadic eye movements. This is due to the fact that individuals sway more in the AP direction than in the ML direction and that there are many more degrees of liberty in the frontal plane than in the sagittal one (e.g. rotation of the knee only in the frontal plane, rotation at the hip and at the level of the vertebrae column larger forward/backward than on the side …). Furthermore, [START_REF] Fukushima | Postural control during downward sway head movements in young subjects[END_REF] explained that when the head moves in an up-down motion (rotation in the pitch direction), the center of mass of the head is moved further away from its position than when the head turns in a left-right motion (rotation in the yaw direction). As the head is a heavy body segment, turning the head in the pitch direction automatically leads the body to sway more than turning the head in the yaw direction does. In fact, with similar amplitudes of head rotations vertically and horizontally, the center of mass of the head is further displaced in vertical as opposed to horizontal saccadic eye movements. As postural sway needs to be reduced to succeed in gaze shift tasks, postural sway should be more carefully controlled, that is, postural control requires further improvement to perform vertical eye movements. This postural control strategy would prevent any increase in postural sway caused by vertical saccadic eye movements. This explains why attention dedicated to postural control has already been found to be higher in vertical saccadic eye movements [START_REF] Tzelepi | Cortical activity preceding vertical saccades: a MEG study[END_REF].

Furthermore, the sway frequency results indicate possible reasons for sway amplitude reduction when the participants performed horizontal and vertical eye movements (Fig 4). In fact, we observed that sway frequency was increased in the AP direction in both eye movement conditions (Fig 4), indicating stronger postural control in the AP axis [START_REF] Winter | Stiffness Control of Balance in Quiet Standing[END_REF]). Hence, this result is in line with a reduction in AP postural sway in the saccadic vertical and horizontal eye movement conditions. Overall, our results led us to advance knowledge of the use of visual information, regarding the horizontal and vertical saccades, in postural control.

Finally, postural sway was always larger for the head than for the trunk (Fig 2 and3), despite both segments moving coherent and strongly related to each other (Table 1) in all eye movement conditions. These results thus validate our second hypothesis showing that even in performing saccadic eye movements of 11°, the body can mimic an inverted pendulum with the ankle most likely functioning as the pivot [START_REF] Winter | Stiffness Control of Balance in Quiet Standing[END_REF][START_REF] Karlsson | Correlations between force plate measures for assessment of balance[END_REF]. This conclusion is further strengthened by the fact that horizontal and vertical eye movements reduce the sway of the whole body, trunk and head in this study. This result is important because the head could have been stabilized in space by moving all body segments in various ways; for example, around the trunk using ankle and hip strategies [START_REF] Creath | A unified view of quiet and perturbed stance: simultaneous co-existing excitable modes[END_REF]. However, our Pearson correlation analyses showed that healthy young adults preferred swaying as an inverted pendulum in all vision conditions employed in this study. In other words, we observed that guided eye movement tasks involving small displays (up to 11 °) do not lead to any changes in how trunk and head are related to each other as an individual sways in the anterior-posterior and medial-lateral directions.

In summary, this study showed that postural sway in the upright stance was stabilized to successfully perform both horizontal and vertical guided eye movement tasks. Most importantly, postural sway was further stabilized for vertical eye movements compared to horizontal eye movements. The reduction of postural sway has practical implication for the performance of standing tasks, as previously indicated [START_REF] Zemková | Physiological Mechanisms of Post-Exercise Balance Impairment[END_REF], and the use of vertical eye movements in daily activities should be further examined in future studies with new visual environments.

Figure Legends

Fig. 1 Representative time series of gaze position for both horizontal (left panels) and vertical (right panels) directions in all three visual conditions: fixation (a, b), horizontal (c, d), and vertical (e, f) guided eye movements Fig. 2 Mean and standard error for sway area for both the trunk and head in all three visual conditions (fixation, horizontal, and vertical guided eye movements). Note: * denotes a significant difference Fig. 3 Mean and standard error for median sway amplitude in the anterior-posterior (a) and mediallateral (b) directions for both trunk and head in all three visual conditions (fixation, horizontal, and vertical guided eye movements). Note: * denotes a significant difference Fig. 4 Mean and standard error for median sway frequency in the anterior-posterior (a) and mediallateral (b) directions for both the trunk and head in all three visual conditions (fixation, horizontal, and vertical guided movements). Note: * denotes a significant difference Table 1 Mean and standard deviation correlation coefficients between the trunk and the head in all three conditions for both anterior-posterior and medial-lateral directions.

Conditions Directions

Fixation

Availability of data and material:

The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request. Code availability: Not applicable Ethics approval: All procedures were approved by the Institutional Ethics Committee, Institute of Biosciences, São Paulo State University.

Declarations

Funding: No funds, grants, or other support was received.

Conflict of interest:

The authors have no conflicts of interest to declare that are relevant to the content of this article. Consent to participate: Prior to the experimental procedures, the participants signed a written consent form Consent to publish: Participants signed informed consent regarding publishing their data.