Continued -fractions with Pisot unit base in

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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 ((π’™βˆ’πŸ )) 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 πΉπ‘ž . 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 RΓ©nyi [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 π‘₯ π‘₯ ∈ [0,1] as π‘₯ = βˆ‘π‘˜β‰₯1 π›½π‘˜π‘˜ , where π‘₯π‘˜ ∈ {0, β‹― , [𝛽]}. The sequence (π‘₯π‘˜ )π‘˜β‰₯1 is called an expansion of π‘₯ in 𝛽 base. There is no expansion uniqueness but, among them, the greatest sequence for the lexicographical order is called the 𝛽-expansion of π‘₯and it is denoted by 𝑑𝛽 (π‘₯ ). The 𝛽-expansion of π‘₯ is constructed by the greedy following algorithm. We consider the 𝛽transformation 𝑇𝛽 : [0,1] β†’ [0,1], π‘₯ β†’ {𝛽π‘₯ } = 𝛽π‘₯ βˆ’ [𝛽π‘₯] and then we define (π‘₯π‘˜ )π‘˜β‰₯1 = 𝑑𝛽 (π‘₯ ) ≔ π‘₯1 π‘₯2 π‘₯3 β‹―, where π‘₯π‘˜ = [π›½π‘‡π›½π‘˜βˆ’1 (π‘₯) ]. π‘₯ 𝑦 In the case π‘₯ β‰₯ 1, there exists a unique integer 𝑖 such that 𝛽 π‘–βˆ’1 ≀ π‘₯ < 𝛽 𝑖 . So one can write 𝑖 = βˆ‘π‘˜β‰₯1 π‘˜π‘˜, 𝛽

π‘₯

𝛽

where (π‘¦π‘˜ )π‘˜β‰₯1 is the 𝛽-expansion of 𝛽𝑖 . Thus, we have ∞

π‘₯ = βˆ‘ π‘₯π‘˜ 𝛽 βˆ’π‘˜ π‘˜=βˆ’π‘›

π‘ π‘’π‘β„Ž π‘‘β„Žπ‘Žπ‘‘ π‘₯π‘˜ = π‘¦π‘˜βˆ’π‘› .

The 𝛽-integer part of π‘₯ is [π‘₯ ]𝛽 = βˆ‘βˆžπ‘˜=βˆ’π‘› π‘₯π‘˜ 𝛽 βˆ’π‘˜ and the 𝛽-fractional part of π‘₯is {π‘₯ }𝛽 = βˆ‘π‘˜>0 π‘₯π‘˜ 𝛽 βˆ’π‘˜ . When {π‘₯ }𝛽 = 0, we denote by ℀𝛽 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 πœ™-fraction of π‘₯ is finite if and only if π‘₯ ∈ β„š(πœ™). In [4], we have studied the continued 𝛽-fraction with formal power series over finite fields and we have characterize elements of π”½π‘ž ((π‘₯ βˆ’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 π”½π‘ž ((π‘₯ βˆ’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 ((π’™βˆ’πŸ ))

Fields of Formal series 𝔽𝒒 ((π’™βˆ’πŸ )) Let π”½π‘ž be the field with π‘ž elements, π”½π‘ž [π‘₯] the ring of polynomials with coefficient in π”½π‘ž , π”½π‘ž (π‘₯) the field of rational functions, π”½π‘ž (π‘₯, 𝛽) the minimal extension of π”½π‘ž containing π‘₯ and 𝛽 and by π”½π‘ž [π‘₯, 𝛽] the minimal ring containing π‘₯ and 𝛽. Let π”½π‘ž ((π‘₯ βˆ’1 )) be the field of formal power series of the form: 𝑙

𝑓=βˆ‘ π‘˜=βˆ’βˆž

where

π‘“π‘˜ π‘₯ π‘˜ ,

π‘“π‘˜ ∈ π”½π‘ž ,

max{π‘˜: π‘“π‘˜ β‰  0} for 𝑙 = deg(𝑓) ≔ { βˆ’βˆž for

𝑓 β‰  0; 𝑓 = 0.

deg(𝑓 ) for 𝑓 β‰  0; |𝑓 | = { π‘ž 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| β‰  |𝑔|. βˆ’1 Let ∈ π”½π‘ž ((π‘₯ )) , the polynomial part of 𝑓 is [𝑓] = βˆ‘π‘˜β‰₯0 π‘“π‘˜ π‘₯ π‘˜ . We know that the empty sum is always equal zero. Therefore, the fraction part is [𝑓] ∈ π”½π‘ž [π‘₯] and {𝑓} = 𝑓 βˆ’ [𝑓] is in the unit disk 𝐷(0,1).

Define the absolute value

An element 𝛽 ∈ π”½π‘ž ((π‘₯ βˆ’1 )) is called a Pisot element if it is an algebraic integer over π”½π‘ž [π‘₯], [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 π”½π‘ž ((π‘₯ βˆ’1 )) : Theorem 2.1.Let 𝛽 ∈ π”½π‘ž ((π‘₯ βˆ’1 ))be an algebraic integer over π”½π‘ž [π‘₯] with the minimal polynomial 𝑃(𝑦) = 𝑦 𝑛 βˆ’ 𝐴1 𝑦 π‘›βˆ’1 βˆ’ β‹― βˆ’ 𝐴𝑛 , 𝐴𝑖 ∈ π”½π‘ž [π‘₯ ]. Then, 𝛽 is a Pisot elements if and only if |𝐴1 | > max |𝐴𝑖 |. 2≀i≀n

Let 𝛽 ∈ π”½π‘ž ((π‘₯ βˆ’1 )) with |𝛽 | > 1. A 𝛽-representation of 𝑓 is an infinite sequences(𝑑𝑖 )𝑖β‰₯1 , where 𝑑𝑖 ∈ 𝑑 π”½π‘ž [π‘₯ ] and 𝑓 = βˆ‘π‘–β‰₯1 𝛽𝑖𝑖 . A 𝛽-expansion of 𝑓, denoted 𝑑𝛽 (𝑓) = (𝑑𝑖 )𝑖β‰₯1 , is a 𝛽-represenation of 𝑓 such that: 𝑑𝑖 = [π›½π‘‡π›½π‘–βˆ’1 (𝑓)] where𝑇𝛽 : 𝐷(0,1) β†’ 𝐷(0,1) 𝑓 β†’ 𝛽𝑓 βˆ’ [𝛽𝑓] . (1) The 𝛽-expansion can be computed by the following algorithm: π‘Ÿ0 = 𝑓 and for 𝑖 β‰₯ 1 𝑑𝑖 = [π›½π‘Ÿπ‘–βˆ’1 ], π‘Ÿπ‘– = π›½π‘Ÿπ‘–βˆ’1 βˆ’ 𝑑𝑖 . The 𝛽-expansion𝑑𝛽 (𝑓) is finite if and only if there is π‘˜ β‰₯ 0 such that 𝑑𝑖 = 0 for all 𝑖 β‰₯ π‘˜. It is called ultimately periodic if and only if there is some smallest 𝑝 β‰₯ 0 (the pre-period length) and 𝑠 β‰₯ 1 (the period length) for which 𝑑𝑖+𝑠 = 𝑑𝑖 for all 𝑖 β‰₯ 𝑝 + 1. Using the last notion, let: 𝐹𝑖𝑛(𝛽 ) = {𝑓 ∈ π”½π‘ž ((π‘₯ βˆ’1 )): 𝑑𝛽 (𝑓)is finite} and π‘ƒπ‘’π‘Ÿ(𝛽 ) = {𝑓 ∈ π”½π‘ž ((π‘₯ βˆ’1 )): 𝑑𝛽 (𝑓)is eventuallly periodic}. When 𝑑𝛽 (𝑓) = 𝑑1 𝑑2 β‹― 𝑑𝑙+1 Β· 𝑑𝑙+2 β‹― π‘‘π‘š then, we denote by deg(𝑓)𝛽 = 𝑙 and ord(𝑓) = π‘š. 𝑓

For |𝑓| β‰₯ 1, then there is a unique π‘˜ ∈ 𝑁 such that |𝛽 |π‘˜ ≀ |𝑓| ≀ |𝛽 |π‘˜+1 . So we have |π›½π‘˜+1 | < 1 and we 𝑓

can represent 𝑓 by shifting 𝑑𝛽 (π›½π‘˜+1 ) by π‘˜ digits to the left. Thus, if 𝑑𝛽 (𝑓) = 0. 𝑑1 𝑑2 β‹― , then𝑑𝛽 (𝛽𝑓) = 𝑑1 . 𝑑2 𝑑3 β‹― Remark 2.1. There is no carry occurring, when we add two polynomials in π”½π‘ž [π‘₯] with degree less than deg(𝛽 ). Consequently, if 𝑓, 𝑔 ∈ π”½π‘ž ((π‘₯ βˆ’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 ) = π”½π‘ž [π‘₯, 𝛽 βˆ’1 ]. Β© 2017, AJMS. All Rights Reserved.

231

Kammoun Rania et al.\ Continued 𝜷-fractions with Pisot unit base in 𝑭𝒒 ((π’™βˆ’πŸ )) π‘™βˆ’π‘–+1 Let 𝑓 ∈ π”½π‘ž ((π‘₯ βˆ’1 )), the 𝛽-polynomial part of 𝑓 is [𝑓]𝛽 = βˆ‘π‘™+1 and the 𝛽-fractional part is 𝑖=1 𝑑𝑖 𝛽 π‘™βˆ’π‘–+1 {𝑓} 𝛽 = 𝑓 βˆ’ [𝑓]𝛽 = βˆ‘π‘–>𝑙+1 𝑑𝑖 𝛽 . We define the set of 𝛽-polynomials as follows: π”½π‘ž [π‘₯]𝛽 = {𝑓 ∈ π”½π‘ž ((π‘₯ βˆ’1 )); {𝑓} 𝛽 = 0}. Then, clearly π”½π‘ž [π‘₯ ]𝛽 βŠ† π”½π‘ž [π‘₯, 𝛽 ]. Furthermore, we introduce the following set β€²

(π”½π‘ž [π‘₯]) = {𝑃 ∈ π”½π‘ž [π‘₯ ]𝜷 , deg(𝑃) ≀ deg(𝛽 ) βˆ’ 1} = {𝑃 ∈ π”½π‘ž [π‘₯ ]𝜷 , 𝑑𝑒𝑔𝛽 (𝑃) = 0}. The set of power series that can be written as a fraction of two 𝛽-polynomials denoted by π”½π‘ž (π‘₯ )𝛽 . Then, clearly π”½π‘ž [π‘₯]𝛽 βŠ† π”½π‘ž (π‘₯, 𝛽). In [3], the authors studied the quantity πΏβŠ™ and they define as follows: πΏβŠ™ = min{𝑛 ∈ β„•: βˆ€ 𝑃1 , 𝑃2 ∈ π”½π‘ž [π‘₯ ]𝛽 ; 𝑃1 𝑃2 ∈ 𝐹𝑖𝑛(𝛽 ) β‡’ 𝛽 𝑛 (𝑃1 𝑃2 ) ∈ π”½π‘ž [π‘₯]𝛽 }. Theorem 2.3.[3] Let 𝛽be a quadratic Pisot unit series. Then πΏβŠ™ = 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 ∈ π”½π‘ž ((π‘₯ βˆ’1 )) with |𝛽 | > 1.When 𝛽 = π‘₯, 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) β†’ 𝐷(0,1) 1

1

𝑓 β†’ 𝑓 βˆ’ [𝑓 ] . 𝛽

when |𝑓| < 1, we obtain 1

𝑓= 𝐴1 +

= [0, 𝐴1 , 𝐴2 , β‹― ]𝛽

1 1 𝐴2 + β‹±

whither (π΄π‘˜ )π‘˜β‰₯1 ∈ π”½π‘ž [π‘₯]𝛽 and there are defined by π΄π‘˜ = [

1

π‘˜βˆ’1 𝑇 ′𝛽 (𝑓)

] , βˆ€ π‘˜ β‰₯ 1. 𝛽

For 𝑓 ∈ π”½π‘ž ((π‘₯ βˆ’1 )) and 𝐴0 = [𝑓]𝛽 , we get 𝑓 = 𝐴0 +

1 𝐴1 +

1

= [𝐴0 , 𝐴1 , 𝐴2 , β‹― ]𝛽 .

1 𝐴2 + β‹± The last bracket is called continued 𝛽-fraction expansion of 𝑓. The sequence (π΄π‘˜ )π‘˜β‰₯0 is called the sequence of partial 𝛽-quotients of 𝑓. We define the π‘›π‘‘β„Ž 𝛽-complete quotient of 𝑓 by 𝑓𝑛 = [𝐴0 , 𝐴1 , 𝐴2 , β‹― , 𝑓𝑛 ]𝛽 . We remark that all (π΄π‘˜ )π‘˜β‰₯1 are not in π”½π‘ž . 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 π”½π‘ž such that deg(𝛽 ) = π‘š. Let 𝛽 ∈ π”½π‘ž (π‘₯, 𝛽) such that the continued 𝛽-fraction of 𝑓 is given by 𝑓 = [𝐴0 , 𝐴1 , 𝐴2 , β‹― , 𝐴𝑛 , β‹― ]. If 𝑓 ∈ π”½π‘ž (π‘₯, 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 π”½π‘ž (π‘₯, 𝛽 ) = π”½π‘ž (π‘₯ )𝛽 . Now, we define two sequences (𝑃𝑛 )π‘›βˆˆβ„• and (𝑄𝑛 )π‘›βˆˆβ„• in π”½π‘ž [π‘₯, 𝛽] by 𝑃 = π‘Ž0 , 𝑃1 = π‘Ž0 π‘Ž1 + 1 𝑃𝑛 = π‘Žπ‘› π‘ƒπ‘›βˆ’1 + π‘ƒπ‘›βˆ’2 { 0 and { 𝑄0 = 1 , 𝑄1 = π‘Ž1 𝑄𝑛 = π‘Žπ‘› π‘„π‘›βˆ’1 + π‘„π‘›βˆ’2 , βˆ€π‘› β‰₯ 2 The pair (𝑃𝑛 , 𝑄𝑛 ) is called reduced 𝛽-fractionary expansion of 𝑓 for all 𝑛 β‰₯ 0. 𝑃 1 Proposition 4.1. Let 𝑓 ∈ π”½π‘ž (π‘₯, 𝛽 ) such that 𝑓 = [𝐴0 , 𝐴1 , 𝐴2 , β‹― , 𝐴𝑛 , β‹― ]. Then |𝑓 βˆ’ 𝑄𝑛 | < |𝑄 |2 . 𝑛

𝑛

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

232

Kammoun Rania et al.\ Continued 𝜷-fractions with Pisot unit base in 𝑭𝒒 ((π’™βˆ’πŸ ))

Proposition 4.2. Let 𝛽 be a quadratic Pisot unit power formal series such that deg(𝛽 ) = π‘š and 𝑃1 , 𝑃2 , β‹― , π‘ƒπ‘š ∈ π”½π‘ž [π‘₯ ]𝛽 . Then, 𝛽 π‘šβˆ’1 𝑃1 𝑃2 β‹― π‘ƒπ‘š ∈ π”½π‘ž [π‘₯ ]𝛽 . The proof of the last proposition is an immediate consequence of Thoerem 2.3. Corollary 4.3. Let 𝑃1 , 𝑃2 , β‹― , π‘ƒπ‘š ∈ π”½π‘ž [π‘₯]𝛽 . Then we have, for all positive integer 𝑛, 𝛽

(π‘šβˆ’1)𝑛 π‘š

𝑃1 𝑃2 β‹― π‘ƒπ‘š ∈ π”½π‘ž [π‘₯]𝛽 .

Corollary 4.4.Let (𝑃𝑛 , 𝑄𝑛 )𝑛β‰₯0 the reduced 𝛽-fractionary expansion of 𝑓 . Then 𝛽

(π‘šβˆ’1)𝑛 π‘š

𝑃𝑛 ∈ π”½π‘ž [π‘₯]𝛽 and

(π‘šβˆ’1)𝑛 π‘š

𝑄𝑛 ∈ π”½π‘ž [π‘₯ ]𝛽 . For 𝑃 = π‘Žπ‘  𝛽 𝑠 + β‹― + π‘Ž0 ∈ π”½π‘ž [π‘₯ ]𝛽 . We denote by 𝛾(𝑃) = π‘š deg 𝛽 (𝑃) + deg 𝑠 (π‘Žπ‘  ) = 2π‘š + deg(π‘Žπ‘  ). Lemma 4.5. [4] Let 𝐴, 𝐡 ∈ π”½π‘ž [π‘₯ ]𝛽 with 𝛾 (𝐴) > 𝛾(𝐡). Then there exists 𝐢, 𝐴1 and 𝐡1 in π”½π‘ž [π‘₯]𝛽 , such that 𝐴 1 =𝐢+ with 𝛾(𝐴1 ) > 𝛾 (𝐡1 ). 𝐴1 𝐡 𝐡1 𝛽

Proof of Theorem 4.1 𝑃 It is equivalent to prove that there exist 𝑛0 β‰₯ 1, 𝐴𝑛 ∈ (π”½π‘ž [π‘₯]𝛽 ) β€². By Lemma 4.2, we obtain 𝑓 = ∈ 𝑄

π”½π‘ž (π‘₯)𝛽 such as 𝑃, 𝑄 ∈ π”½π‘ž [π‘₯]𝛽 and (𝑃𝑛 , 𝑄𝑛 ) the reduced 𝛽-fractionary expansion of 𝑓. 𝑃

𝑃

1

By proposition 4.1, |𝑄 βˆ’ 𝑄𝑛 | < |𝑄 𝑛

(𝛽

(π‘šβˆ’1) (𝑛+1) π‘š

2 𝑛|

. According to Corollary 4.3 and Lemma 4.5, we have

(𝑃𝑄𝑛 βˆ’ 𝑄𝑃𝑛 )) inπ”½π‘ž [π‘₯]𝛽 .So, we obtain (π‘šβˆ’1)(𝑛+1)

π‘š 1 𝑃𝑄𝑛 βˆ’ 𝑄𝑃𝑛 |𝛽| < |𝛽 |(π‘šβˆ’1)(𝑛+1)/π‘š | |< |𝑄| 𝑄 |𝑄𝑛 | which implies that deg(𝑄𝑛 ) ≀ deg(𝑄) + (π‘š βˆ’ 1)(𝑛 + 1), where 𝑛

deg(𝑄𝑛 ) = βˆ‘ deg(𝐴𝑖 ) ≀ deg(𝑄) + (π‘š βˆ’ 1)(𝑛 + 1). 𝑖=1

βˆ‘π‘›π‘–=1(deg(𝐴𝑖 ) βˆ’ (π‘š βˆ’ 1)) ≀ deg(𝑄) + (π‘š βˆ’ 1). Finally there exists 𝑛0 β‰₯ 1, such that, for Thus deg(𝐴𝑖 ) βˆ’ (π‘š βˆ’ 1) ≀ 0, for all 𝑖 β‰₯ 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 πΉπ‘ž ((π‘₯ βˆ’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. RΓ©nyi. 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.

Β© 2017, AJMS. All Rights Reserved.

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