Continued -fractions with Pisot unit base in

Available Online at www.ajms.in Asian Journal of Mathematical Sciences 2017; 1(6):230-233 RESEARCH ARTICLE

Continued 𝜷-fractions with Pisot unit base in 𝑭𝒒 ((𝒙−𝟏 )) Rania Kammoun* * University of Sfax, Faculty of Sciences, Department of Mathematics, Algebra Laboratory, Geometry and Spectral Theory (AGTS) LR11ES53, BP 802, 3038 Sfax, Tunisia.

Receivedon:15/11/2017,Revisedon:01/12/2017,Acceptedon:29/12/2017 ABSTRACT In this paper, we are interested in introducing a new theory of continued fractions based on the betaexpansion theory in the field of Laurent series over a finite field 𝐹𝑞 . We will characterize all elements having finite continued beta-fraction where the base is a unit Pisot quadratic series. Classification Mathematic Subject: 11R06, 37B50. Key words: Continued 𝛽-fraction, formal power series, Pisot series, 𝛽-expansion, finite field. INTRODUCTION The 𝛽-numeration introduced in 1957 by Rényi [5] is a new numeration system when we replace the integer base b with a non-integral base. Let 𝛽 > 1, in the case of a non-integral base, one may write any 𝑥 𝑥 ∈ [0,1] as 𝑥 = ∑𝑘≥1 𝛽𝑘𝑘 , where 𝑥𝑘 ∈ {0, ⋯ , [𝛽]}. The sequence (𝑥𝑘 )𝑘≥1 is called an expansion of 𝑥 in 𝛽 base. There is no expansion uniqueness but, among them, the greatest sequence for the lexicographical order is called the 𝛽-expansion of 𝑥and it is denoted by 𝑑𝛽 (𝑥 ). The 𝛽-expansion of 𝑥 is constructed by the greedy following algorithm. We consider the 𝛽transformation 𝑇𝛽 : [0,1] → [0,1], 𝑥 → {𝛽𝑥 } = 𝛽𝑥 − [𝛽𝑥] and then we define (𝑥𝑘 )𝑘≥1 = 𝑑𝛽 (𝑥 ) ≔ 𝑥1 𝑥2 𝑥3 ⋯, where 𝑥𝑘 = [𝛽𝑇𝛽𝑘−1 (𝑥) ]. 𝑥 𝑦 In the case 𝑥 ≥ 1, there exists a unique integer 𝑖 such that 𝛽 𝑖−1 ≤ 𝑥 < 𝛽 𝑖 . So one can write 𝑖 = ∑𝑘≥1 𝑘𝑘, 𝛽

𝑥

𝛽

where (𝑦𝑘 )𝑘≥1 is the 𝛽-expansion of 𝛽𝑖 . Thus, we have ∞

𝑥 = ∑ 𝑥𝑘 𝛽 −𝑘 𝑘=−𝑛

𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 𝑥𝑘 = 𝑦𝑘−𝑛 .

The 𝛽-integer part of 𝑥 is [𝑥 ]𝛽 = ∑∞𝑘=−𝑛 𝑥𝑘 𝛽 −𝑘 and the 𝛽-fractional part of 𝑥is {𝑥 }𝛽 = ∑𝑘>0 𝑥𝑘 𝛽 −𝑘 . When {𝑥 }𝛽 = 0, we denote by ℤ𝛽 the set of all 𝛽-integers. Obviously, we can present an algorithm of continued fractions similarly to the classical decimal case by consideration 𝛽 ∈ ℝ (non-integer) and then we get the so called continued 𝛽-fraction, whither the sequence of partial quotients consists of 𝛽-integers instead of integers. In [2], J. Bernat has showed that the continued 𝜙-fraction of 𝑥 is finite if and only if 𝑥 ∈ ℚ(𝜙). In [4], we have studied the continued 𝛽-fraction with formal power series over finite fields and we have characterize elements of 𝔽𝑞 ((𝑥 −1 )) having finite 𝛽-fraction when the base 𝛽 is a quadratic Pisot unit. Throughout this paper, we improve the result given [4] by studying the case when 𝛽 is only a Pisot unit in 𝔽𝑞 ((𝑥 −1 )). The paper is organized as follows, Section 2, we introduce some basic definitions and results. In Section 3, we define the continued 𝛽-fraction expansion. In Section 4, we state our main result. *Corresponding Author: Rania Kammoun, Email: [email protected]

Kammoun Rania et al.\ Continued 𝜷-fractions with Pisot unit base in 𝑭𝒒 ((𝒙−𝟏 ))

Fields of Formal series 𝔽𝒒 ((𝒙−𝟏 )) Let 𝔽𝑞 be the field with 𝑞 elements, 𝔽𝑞 [𝑥] the ring of polynomials with coefficient in 𝔽𝑞 , 𝔽𝑞 (𝑥) the field of rational functions, 𝔽𝑞 (𝑥, 𝛽) the minimal extension of 𝔽𝑞 containing 𝑥 and 𝛽 and by 𝔽𝑞 [𝑥, 𝛽] the minimal ring containing 𝑥 and 𝛽. Let 𝔽𝑞 ((𝑥 −1 )) be the field of formal power series of the form: 𝑙

𝑓=∑ 𝑘=−∞

where

𝑓𝑘 𝑥 𝑘 ,

𝑓𝑘 ∈ 𝔽𝑞 ,

max{𝑘: 𝑓𝑘 ≠ 0} for 𝑙 = deg(𝑓) ≔ { −∞ for

𝑓 ≠ 0; 𝑓 = 0.

deg(𝑓 ) for 𝑓 ≠ 0; |𝑓 | = { 𝑞 0 for 𝑓 = 0. As |. | is not Archimedean, it satisfies the strict triangle inequality |𝑓 + 𝑔| ≤ max(|𝑓|, |𝑔|) and |𝑓 + 𝑔| = max(|𝑓|, |𝑔|) if |𝑓| ≠ |𝑔|. −1 Let ∈ 𝔽𝑞 ((𝑥 )) , the polynomial part of 𝑓 is [𝑓] = ∑𝑘≥0 𝑓𝑘 𝑥 𝑘 . We know that the empty sum is always equal zero. Therefore, the fraction part is [𝑓] ∈ 𝔽𝑞 [𝑥] and {𝑓} = 𝑓 − [𝑓] is in the unit disk 𝐷(0,1).

Define the absolute value

An element 𝛽 ∈ 𝔽𝑞 ((𝑥 −1 )) is called a Pisot element if it is an algebraic integer over 𝔽𝑞 [𝑥], [𝛽 ] > 1 and|𝛽𝑖 | < 1 for all conjugates 𝛽. Using the coefficient of minimal polynomial, P. Batman and A.L. Duquette [1] had characterized the Pisot elements in 𝔽𝑞 ((𝑥 −1 )) : Theorem 2.1.Let 𝛽 ∈ 𝔽𝑞 ((𝑥 −1 ))be an algebraic integer over 𝔽𝑞 [𝑥] with the minimal polynomial 𝑃(𝑦) = 𝑦 𝑛 − 𝐴1 𝑦 𝑛−1 − ⋯ − 𝐴𝑛 , 𝐴𝑖 ∈ 𝔽𝑞 [𝑥 ]. Then, 𝛽 is a Pisot elements if and only if |𝐴1 | > max |𝐴𝑖 |. 2≤i≤n

Let 𝛽 ∈ 𝔽𝑞 ((𝑥 −1 )) with |𝛽 | > 1. A 𝛽-representation of 𝑓 is an infinite sequences(𝑑𝑖 )𝑖≥1 , where 𝑑𝑖 ∈ 𝑑 𝔽𝑞 [𝑥 ] and 𝑓 = ∑𝑖≥1 𝛽𝑖𝑖 . A 𝛽-expansion of 𝑓, denoted 𝑑𝛽 (𝑓) = (𝑑𝑖 )𝑖≥1 , is a 𝛽-represenation of 𝑓 such that: 𝑑𝑖 = [𝛽𝑇𝛽𝑖−1 (𝑓)] where𝑇𝛽 : 𝐷(0,1) → 𝐷(0,1) 𝑓 → 𝛽𝑓 − [𝛽𝑓] . (1) The 𝛽-expansion can be computed by the following algorithm: 𝑟0 = 𝑓 and for 𝑖 ≥ 1 𝑑𝑖 = [𝛽𝑟𝑖−1 ], 𝑟𝑖 = 𝛽𝑟𝑖−1 − 𝑑𝑖 . The 𝛽-expansion𝑑𝛽 (𝑓) is finite if and only if there is 𝑘 ≥ 0 such that 𝑑𝑖 = 0 for all 𝑖 ≥ 𝑘. It is called ultimately periodic if and only if there is some smallest 𝑝 ≥ 0 (the pre-period length) and 𝑠 ≥ 1 (the period length) for which 𝑑𝑖+𝑠 = 𝑑𝑖 for all 𝑖 ≥ 𝑝 + 1. Using the last notion, let: 𝐹𝑖𝑛(𝛽 ) = {𝑓 ∈ 𝔽𝑞 ((𝑥 −1 )): 𝑑𝛽 (𝑓)is finite} and 𝑃𝑒𝑟(𝛽 ) = {𝑓 ∈ 𝔽𝑞 ((𝑥 −1 )): 𝑑𝛽 (𝑓)is eventuallly periodic}. When 𝑑𝛽 (𝑓) = 𝑑1 𝑑2 ⋯ 𝑑𝑙+1 · 𝑑𝑙+2 ⋯ 𝑑𝑚 then, we denote by deg(𝑓)𝛽 = 𝑙 and ord(𝑓) = 𝑚. 𝑓

For |𝑓| ≥ 1, then there is a unique 𝑘 ∈ 𝑁 such that |𝛽 |𝑘 ≤ |𝑓| ≤ |𝛽 |𝑘+1 . So we have |𝛽𝑘+1 | < 1 and we 𝑓

can represent 𝑓 by shifting 𝑑𝛽 (𝛽𝑘+1 ) by 𝑘 digits to the left. Thus, if 𝑑𝛽 (𝑓) = 0. 𝑑1 𝑑2 ⋯ , then𝑑𝛽 (𝛽𝑓) = 𝑑1 . 𝑑2 𝑑3 ⋯ Remark 2.1. There is no carry occurring, when we add two polynomials in 𝔽𝑞 [𝑥] with degree less than deg(𝛽 ). Consequently, if 𝑓, 𝑔 ∈ 𝔽𝑞 ((𝑥 −1 )), we get 𝑑𝛽 (𝑓 + 𝑔) = 𝑑𝛽 (𝑓) + 𝑑𝛽 (𝑔). In [6], Scheicher has characterized the set 𝐹𝑖𝑛 (𝛽) when 𝛽 is Pisot. Theorem 2.2.[6] 𝛽 is a Pisot series if and only if 𝐹𝑖𝑛(𝛽 ) = 𝔽𝑞 [𝑥, 𝛽 −1 ]. © 2017, AJMS. All Rights Reserved.

231

Kammoun Rania et al.\ Continued 𝜷-fractions with Pisot unit base in 𝑭𝒒 ((𝒙−𝟏 )) 𝑙−𝑖+1 Let 𝑓 ∈ 𝔽𝑞 ((𝑥 −1 )), the 𝛽-polynomial part of 𝑓 is [𝑓]𝛽 = ∑𝑙+1 and the 𝛽-fractional part is 𝑖=1 𝑑𝑖 𝛽 𝑙−𝑖+1 {𝑓} 𝛽 = 𝑓 − [𝑓]𝛽 = ∑𝑖>𝑙+1 𝑑𝑖 𝛽 . We define the set of 𝛽-polynomials as follows: 𝔽𝑞 [𝑥]𝛽 = {𝑓 ∈ 𝔽𝑞 ((𝑥 −1 )); {𝑓} 𝛽 = 0}. Then, clearly 𝔽𝑞 [𝑥 ]𝛽 ⊆ 𝔽𝑞 [𝑥, 𝛽 ]. Furthermore, we introduce the following set ′

(𝔽𝑞 [𝑥]) = {𝑃 ∈ 𝔽𝑞 [𝑥 ]𝜷 , deg(𝑃) ≤ deg(𝛽 ) − 1} = {𝑃 ∈ 𝔽𝑞 [𝑥 ]𝜷 , 𝑑𝑒𝑔𝛽 (𝑃) = 0}. The set of power series that can be written as a fraction of two 𝛽-polynomials denoted by 𝔽𝑞 (𝑥 )𝛽 . Then, clearly 𝔽𝑞 [𝑥]𝛽 ⊆ 𝔽𝑞 (𝑥, 𝛽). In [3], the authors studied the quantity 𝐿⊙ and they define as follows: 𝐿⊙ = min{𝑛 ∈ ℕ: ∀ 𝑃1 , 𝑃2 ∈ 𝔽𝑞 [𝑥 ]𝛽 ; 𝑃1 𝑃2 ∈ 𝐹𝑖𝑛(𝛽 ) ⇒ 𝛽 𝑛 (𝑃1 𝑃2 ) ∈ 𝔽𝑞 [𝑥]𝛽 }. Theorem 2.3.[3] Let 𝛽be a quadratic Pisot unit series. Then 𝐿⊙ = 1. Continued 𝜷-fraction algorithm We begin by introduce a generalization of the algorithm of the expansion in continued fraction in the field of formal power series in base 𝛽 ∈ 𝔽𝑞 ((𝑥 −1 )) with |𝛽 | > 1.When 𝛽 = 𝑥, this theory is seems to be similar to the classical case of continued fractions. We define the 𝛽-transformation 𝑇𝛽′ by: 𝑇𝛽′ : 𝐷(0,1) → 𝐷(0,1) 1

1

𝑓 → 𝑓 − [𝑓 ] . 𝛽

when |𝑓| < 1, we obtain 1

𝑓= 𝐴1 +

= [0, 𝐴1 , 𝐴2 , ⋯ ]𝛽

1 1 𝐴2 + ⋱

whither (𝐴𝑘 )𝑘≥1 ∈ 𝔽𝑞 [𝑥]𝛽 and there are defined by 𝐴𝑘 = [

1

𝑘−1 𝑇 ′𝛽 (𝑓)

] , ∀ 𝑘 ≥ 1. 𝛽

For 𝑓 ∈ 𝔽𝑞 ((𝑥 −1 )) and 𝐴0 = [𝑓]𝛽 , we get 𝑓 = 𝐴0 +

1 𝐴1 +

1

= [𝐴0 , 𝐴1 , 𝐴2 , ⋯ ]𝛽 .

1 𝐴2 + ⋱ The last bracket is called continued 𝛽-fraction expansion of 𝑓. The sequence (𝐴𝑘 )𝑘≥0 is called the sequence of partial 𝛽-quotients of 𝑓. We define the 𝑛𝑡ℎ 𝛽-complete quotient of 𝑓 by 𝑓𝑛 = [𝐴0 , 𝐴1 , 𝐴2 , ⋯ , 𝑓𝑛 ]𝛽 . We remark that all (𝐴𝑘 )𝑘≥1 are not in 𝔽𝑞 . Main Results Our main result is an improvement of Theorem 4.1 in [4]. Theorem 4.1. Let 𝛽 be a quadratic Pisot unit formal power series over the finite field 𝔽𝑞 such that deg(𝛽 ) = 𝑚. Let 𝛽 ∈ 𝔽𝑞 (𝑥, 𝛽) such that the continued 𝛽-fraction of 𝑓 is given by 𝑓 = [𝐴0 , 𝐴1 , 𝐴2 , ⋯ , 𝐴𝑛 , ⋯ ]. If 𝑓 ∈ 𝔽𝑞 (𝑥, 𝛽) then {𝐴𝑖 / deg 𝛽 (𝐴𝑖 ) > 0} is finite. So as to prove the above Theorem, first we need to recall some results given in [4] and we use the following Lemmas and Propositions. Lemma 4.2. [4] Let 𝛽 be a unit Pisot series. Then 𝔽𝑞 (𝑥, 𝛽 ) = 𝔽𝑞 (𝑥 )𝛽 . Now, we define two sequences (𝑃𝑛 )𝑛∈ℕ and (𝑄𝑛 )𝑛∈ℕ in 𝔽𝑞 [𝑥, 𝛽] by 𝑃 = 𝑎0 , 𝑃1 = 𝑎0 𝑎1 + 1 𝑃𝑛 = 𝑎𝑛 𝑃𝑛−1 + 𝑃𝑛−2 { 0 and { 𝑄0 = 1 , 𝑄1 = 𝑎1 𝑄𝑛 = 𝑎𝑛 𝑄𝑛−1 + 𝑄𝑛−2 , ∀𝑛 ≥ 2 The pair (𝑃𝑛 , 𝑄𝑛 ) is called reduced 𝛽-fractionary expansion of 𝑓 for all 𝑛 ≥ 0. 𝑃 1 Proposition 4.1. Let 𝑓 ∈ 𝔽𝑞 (𝑥, 𝛽 ) such that 𝑓 = [𝐴0 , 𝐴1 , 𝐴2 , ⋯ , 𝐴𝑛 , ⋯ ]. Then |𝑓 − 𝑄𝑛 | < |𝑄 |2 . 𝑛

𝑛

Proof. Similarly to the classical case. © 2017, AJMS. All Rights Reserved.

232

Kammoun Rania et al.\ Continued 𝜷-fractions with Pisot unit base in 𝑭𝒒 ((𝒙−𝟏 ))

Proposition 4.2. Let 𝛽 be a quadratic Pisot unit power formal series such that deg(𝛽 ) = 𝑚 and 𝑃1 , 𝑃2 , ⋯ , 𝑃𝑚 ∈ 𝔽𝑞 [𝑥 ]𝛽 . Then, 𝛽 𝑚−1 𝑃1 𝑃2 ⋯ 𝑃𝑚 ∈ 𝔽𝑞 [𝑥 ]𝛽 . The proof of the last proposition is an immediate consequence of Thoerem 2.3. Corollary 4.3. Let 𝑃1 , 𝑃2 , ⋯ , 𝑃𝑚 ∈ 𝔽𝑞 [𝑥]𝛽 . Then we have, for all positive integer 𝑛, 𝛽

(𝑚−1)𝑛 𝑚

𝑃1 𝑃2 ⋯ 𝑃𝑚 ∈ 𝔽𝑞 [𝑥]𝛽 .

Corollary 4.4.Let (𝑃𝑛 , 𝑄𝑛 )𝑛≥0 the reduced 𝛽-fractionary expansion of 𝑓 . Then 𝛽

(𝑚−1)𝑛 𝑚

𝑃𝑛 ∈ 𝔽𝑞 [𝑥]𝛽 and

(𝑚−1)𝑛 𝑚

𝑄𝑛 ∈ 𝔽𝑞 [𝑥 ]𝛽 . For 𝑃 = 𝑎𝑠 𝛽 𝑠 + ⋯ + 𝑎0 ∈ 𝔽𝑞 [𝑥 ]𝛽 . We denote by 𝛾(𝑃) = 𝑚 deg 𝛽 (𝑃) + deg 𝑠 (𝑎𝑠 ) = 2𝑚 + deg(𝑎𝑠 ). Lemma 4.5. [4] Let 𝐴, 𝐵 ∈ 𝔽𝑞 [𝑥 ]𝛽 with 𝛾 (𝐴) > 𝛾(𝐵). Then there exists 𝐶, 𝐴1 and 𝐵1 in 𝔽𝑞 [𝑥]𝛽 , such that 𝐴 1 =𝐶+ with 𝛾(𝐴1 ) > 𝛾 (𝐵1 ). 𝐴1 𝐵 𝐵1 𝛽

Proof of Theorem 4.1 𝑃 It is equivalent to prove that there exist 𝑛0 ≥ 1, 𝐴𝑛 ∈ (𝔽𝑞 [𝑥]𝛽 ) ′. By Lemma 4.2, we obtain 𝑓 = ∈ 𝑄

𝔽𝑞 (𝑥)𝛽 such as 𝑃, 𝑄 ∈ 𝔽𝑞 [𝑥]𝛽 and (𝑃𝑛 , 𝑄𝑛 ) the reduced 𝛽-fractionary expansion of 𝑓. 𝑃

𝑃

1

By proposition 4.1, |𝑄 − 𝑄𝑛 | < |𝑄 𝑛

(𝛽

(𝑚−1) (𝑛+1) 𝑚

2 𝑛|

. According to Corollary 4.3 and Lemma 4.5, we have

(𝑃𝑄𝑛 − 𝑄𝑃𝑛 )) in𝔽𝑞 [𝑥]𝛽 .So, we obtain (𝑚−1)(𝑛+1)

𝑚 1 𝑃𝑄𝑛 − 𝑄𝑃𝑛 |𝛽| < |𝛽 |(𝑚−1)(𝑛+1)/𝑚 | |< |𝑄| 𝑄 |𝑄𝑛 | which implies that deg(𝑄𝑛 ) ≤ deg(𝑄) + (𝑚 − 1)(𝑛 + 1), where 𝑛

deg(𝑄𝑛 ) = ∑ deg(𝐴𝑖 ) ≤ deg(𝑄) + (𝑚 − 1)(𝑛 + 1). 𝑖=1

∑𝑛𝑖=1(deg(𝐴𝑖 ) − (𝑚 − 1)) ≤ deg(𝑄) + (𝑚 − 1). Finally there exists 𝑛0 ≥ 1, such that, for Thus deg(𝐴𝑖 ) − (𝑚 − 1) ≤ 0, for all 𝑖 ≥ 𝑛0 and the desired result is reached. REFERENCES 1. P. Bateman and L. Duquette. The analogue of Pisot- Vijayaraghvan numbers in fields of power series, Ill. J. Math, 6, (1962), 594-606. 2. J. Bernat. Continued fractions and numeration in the Fibonacci base, Discrete Mathematics, 22, (2006), 2828-2850. 3. R. Ghorbel, M. Hbaib and S. Zouari. Arithmetics on beta-expansions with Pisot bases over 𝐹𝑞 ((𝑥 −1 )), Bull. Belg. Math. Soc. Simon Stevin, 21, (2014), 241-251. 4. M. Hbaib, R. Kammoun. Continued beta-fractions with formal power series over finite fields, Ramaujan J Math, (2015), DOI 10.1007/s11139-015-9725-5. 5. A. Rényi. Representations for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hung, 8, (1957), 477-493. 6. K. Scheicher. Beta-expansions in algebraic function fields over finite fields, finite fields and their Applications, (2007), 394-410.

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