Available Online at www.ajms.in Asian Journal of Mathematical Sciences 2017; 1(6):230-233 RESEARCH ARTICLE
Continued π·-fractions with Pisot unit base in ππ ((πβπ )) 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 πΉπ . 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 π½-fraction, formal power series, Pisot series, π½-expansion, finite field. INTRODUCTION The π½-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 π½ > 1, in the case of a non-integral base, one may write any π₯ π₯ β [0,1] as π₯ = βπβ₯1 π½ππ , where π₯π β {0, β― , [π½]}. The sequence (π₯π )πβ₯1 is called an expansion of π₯ in π½ base. There is no expansion uniqueness but, among them, the greatest sequence for the lexicographical order is called the π½-expansion of π₯and it is denoted by ππ½ (π₯ ). The π½-expansion of π₯ is constructed by the greedy following algorithm. We consider the π½transformation ππ½ : [0,1] β [0,1], π₯ β {π½π₯ } = π½π₯ β [π½π₯] and then we define (π₯π )πβ₯1 = ππ½ (π₯ ) β π₯1 π₯2 π₯3 β―, where π₯π = [π½ππ½πβ1 (π₯) ]. π₯ π¦ In the case π₯ β₯ 1, there exists a unique integer π such that π½ πβ1 β€ π₯ < π½ π . So one can write π = βπβ₯1 ππ, π½
π₯
π½
where (π¦π )πβ₯1 is the π½-expansion of π½π . Thus, we have β
π₯ = β π₯π π½ βπ π=βπ
π π’πβ π‘βππ‘ π₯π = π¦πβπ .
The π½-integer part of π₯ is [π₯ ]π½ = ββπ=βπ π₯π π½ βπ and the π½-fractional part of π₯is {π₯ }π½ = βπ>0 π₯π π½ βπ . When {π₯ }π½ = 0, we denote by β€π½ the set of all π½-integers. Obviously, we can present an algorithm of continued fractions similarly to the classical decimal case by consideration π½ β β (non-integer) and then we get the so called continued π½-fraction, whither the sequence of partial quotients consists of π½-integers instead of integers. In [2], J. Bernat has showed that the continued π-fraction of π₯ is finite if and only if π₯ β β(π). In [4], we have studied the continued π½-fraction with formal power series over finite fields and we have characterize elements of π½π ((π₯ β1 )) having finite π½-fraction when the base π½ is a quadratic Pisot unit. Throughout this paper, we improve the result given [4] by studying the case when π½ is only a Pisot unit in π½π ((π₯ β1 )). The paper is organized as follows, Section 2, we introduce some basic definitions and results. In Section 3, we define the continued π½-fraction expansion. In Section 4, we state our main result. *Corresponding Author: Rania Kammoun, Email:
[email protected]
Kammoun Rania et al.\ Continued π·-fractions with Pisot unit base in ππ ((πβπ ))
Fields of Formal series π½π ((πβπ )) Let π½π be the field with π elements, π½π [π₯] the ring of polynomials with coefficient in π½π , π½π (π₯) the field of rational functions, π½π (π₯, π½) the minimal extension of π½π containing π₯ and π½ and by π½π [π₯, π½] the minimal ring containing π₯ and π½. Let π½π ((π₯ β1 )) be the field of formal power series of the form: π
π=β π=ββ
where
ππ π₯ π ,
ππ β π½π ,
max{π: ππ β 0} for π = deg(π) β { ββ for
π β 0; π = 0.
deg(π ) for π β 0; |π | = { π 0 for π = 0. As |. | is not Archimedean, it satisfies the strict triangle inequality |π + π| β€ max(|π|, |π|) and |π + π| = max(|π|, |π|) if |π| β |π|. β1 Let β π½π ((π₯ )) , the polynomial part of π is [π] = βπβ₯0 ππ π₯ π . We know that the empty sum is always equal zero. Therefore, the fraction part is [π] β π½π [π₯] and {π} = π β [π] is in the unit disk π·(0,1).
Define the absolute value
An element π½ β π½π ((π₯ β1 )) is called a Pisot element if it is an algebraic integer over π½π [π₯], [π½ ] > 1 and|π½π | < 1 for all conjugates π½. Using the coefficient of minimal polynomial, P. Batman and A.L. Duquette [1] had characterized the Pisot elements in π½π ((π₯ β1 )) : Theorem 2.1.Let π½ β π½π ((π₯ β1 ))be an algebraic integer over π½π [π₯] with the minimal polynomial π(π¦) = π¦ π β π΄1 π¦ πβ1 β β― β π΄π , π΄π β π½π [π₯ ]. Then, π½ is a Pisot elements if and only if |π΄1 | > max |π΄π |. 2β€iβ€n
Let π½ β π½π ((π₯ β1 )) with |π½ | > 1. A π½-representation of π is an infinite sequences(ππ )πβ₯1 , where ππ β π π½π [π₯ ] and π = βπβ₯1 π½ππ . A π½-expansion of π, denoted ππ½ (π) = (ππ )πβ₯1 , is a π½-represenation of π such that: ππ = [π½ππ½πβ1 (π)] whereππ½ : π·(0,1) β π·(0,1) π β π½π β [π½π] . (1) The π½-expansion can be computed by the following algorithm: π0 = π and for π β₯ 1 ππ = [π½ππβ1 ], ππ = π½ππβ1 β ππ . The π½-expansionππ½ (π) is finite if and only if there is π β₯ 0 such that ππ = 0 for all π β₯ π. It is called ultimately periodic if and only if there is some smallest π β₯ 0 (the pre-period length) and π β₯ 1 (the period length) for which ππ+π = ππ for all π β₯ π + 1. Using the last notion, let: πΉππ(π½ ) = {π β π½π ((π₯ β1 )): ππ½ (π)is finite} and πππ(π½ ) = {π β π½π ((π₯ β1 )): ππ½ (π)is eventuallly periodic}. When ππ½ (π) = π1 π2 β― ππ+1 Β· ππ+2 β― ππ then, we denote by deg(π)π½ = π and ord(π) = π. π
For |π| β₯ 1, then there is a unique π β π such that |π½ |π β€ |π| β€ |π½ |π+1 . So we have |π½π+1 | < 1 and we π
can represent π by shifting ππ½ (π½π+1 ) by π digits to the left. Thus, if ππ½ (π) = 0. π1 π2 β― , thenππ½ (π½π) = π1 . π2 π3 β― Remark 2.1. There is no carry occurring, when we add two polynomials in π½π [π₯] with degree less than deg(π½ ). Consequently, if π, π β π½π ((π₯ β1 )), we get ππ½ (π + π) = ππ½ (π) + ππ½ (π). In [6], Scheicher has characterized the set πΉππ (π½) when π½ is Pisot. Theorem 2.2.[6] π½ is a Pisot series if and only if πΉππ(π½ ) = π½π [π₯, π½ β1 ]. Β© 2017, AJMS. All Rights Reserved.
231
Kammoun Rania et al.\ Continued π·-fractions with Pisot unit base in ππ ((πβπ )) πβπ+1 Let π β π½π ((π₯ β1 )), the π½-polynomial part of π is [π]π½ = βπ+1 and the π½-fractional part is π=1 ππ π½ πβπ+1 {π} π½ = π β [π]π½ = βπ>π+1 ππ π½ . We define the set of π½-polynomials as follows: π½π [π₯]π½ = {π β π½π ((π₯ β1 )); {π} π½ = 0}. Then, clearly π½π [π₯ ]π½ β π½π [π₯, π½ ]. Furthermore, we introduce the following set β²
(π½π [π₯]) = {π β π½π [π₯ ]π· , deg(π) β€ deg(π½ ) β 1} = {π β π½π [π₯ ]π· , ππππ½ (π) = 0}. The set of power series that can be written as a fraction of two π½-polynomials denoted by π½π (π₯ )π½ . Then, clearly π½π [π₯]π½ β π½π (π₯, π½). In [3], the authors studied the quantity πΏβ and they define as follows: πΏβ = min{π β β: β π1 , π2 β π½π [π₯ ]π½ ; π1 π2 β πΉππ(π½ ) β π½ π (π1 π2 ) β π½π [π₯]π½ }. Theorem 2.3.[3] Let π½be a quadratic Pisot unit series. Then πΏβ = 1. Continued π·-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 π½ β π½π ((π₯ β1 )) with |π½ | > 1.When π½ = π₯, this theory is seems to be similar to the classical case of continued fractions. We define the π½-transformation ππ½β² by: ππ½β² : π·(0,1) β π·(0,1) 1
1
π β π β [π ] . π½
when |π| < 1, we obtain 1
π= π΄1 +
= [0, π΄1 , π΄2 , β― ]π½
1 1 π΄2 + β±
whither (π΄π )πβ₯1 β π½π [π₯]π½ and there are defined by π΄π = [
1
πβ1 π β²π½ (π)
] , β π β₯ 1. π½
For π β π½π ((π₯ β1 )) and π΄0 = [π]π½ , we get π = π΄0 +
1 π΄1 +
1
= [π΄0 , π΄1 , π΄2 , β― ]π½ .
1 π΄2 + β± The last bracket is called continued π½-fraction expansion of π. The sequence (π΄π )πβ₯0 is called the sequence of partial π½-quotients of π. We define the ππ‘β π½-complete quotient of π by ππ = [π΄0 , π΄1 , π΄2 , β― , ππ ]π½ . We remark that all (π΄π )πβ₯1 are not in π½π . Main Results Our main result is an improvement of Theorem 4.1 in [4]. Theorem 4.1. Let π½ be a quadratic Pisot unit formal power series over the finite field π½π such that deg(π½ ) = π. Let π½ β π½π (π₯, π½) such that the continued π½-fraction of π is given by π = [π΄0 , π΄1 , π΄2 , β― , π΄π , β― ]. If π β π½π (π₯, π½) then {π΄π / deg π½ (π΄π ) > 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 π½ be a unit Pisot series. Then π½π (π₯, π½ ) = π½π (π₯ )π½ . Now, we define two sequences (ππ )πββ and (ππ )πββ in π½π [π₯, π½] by π = π0 , π1 = π0 π1 + 1 ππ = ππ ππβ1 + ππβ2 { 0 and { π0 = 1 , π1 = π1 ππ = ππ ππβ1 + ππβ2 , βπ β₯ 2 The pair (ππ , ππ ) is called reduced π½-fractionary expansion of π for all π β₯ 0. π 1 Proposition 4.1. Let π β π½π (π₯, π½ ) such that π = [π΄0 , π΄1 , π΄2 , β― , π΄π , β― ]. Then |π β ππ | < |π |2 . π
π
Proof. Similarly to the classical case. Β© 2017, AJMS. All Rights Reserved.
232
Kammoun Rania et al.\ Continued π·-fractions with Pisot unit base in ππ ((πβπ ))
Proposition 4.2. Let π½ be a quadratic Pisot unit power formal series such that deg(π½ ) = π and π1 , π2 , β― , ππ β π½π [π₯ ]π½ . Then, π½ πβ1 π1 π2 β― ππ β π½π [π₯ ]π½ . The proof of the last proposition is an immediate consequence of Thoerem 2.3. Corollary 4.3. Let π1 , π2 , β― , ππ β π½π [π₯]π½ . Then we have, for all positive integer π, π½
(πβ1)π π
π1 π2 β― ππ β π½π [π₯]π½ .
Corollary 4.4.Let (ππ , ππ )πβ₯0 the reduced π½-fractionary expansion of π . Then π½
(πβ1)π π
ππ β π½π [π₯]π½ and
(πβ1)π π
ππ β π½π [π₯ ]π½ . For π = ππ π½ π + β― + π0 β π½π [π₯ ]π½ . We denote by πΎ(π) = π deg π½ (π) + deg π (ππ ) = 2π + deg(ππ ). Lemma 4.5. [4] Let π΄, π΅ β π½π [π₯ ]π½ with πΎ (π΄) > πΎ(π΅). Then there exists πΆ, π΄1 and π΅1 in π½π [π₯]π½ , such that π΄ 1 =πΆ+ with πΎ(π΄1 ) > πΎ (π΅1 ). π΄1 π΅ π΅1 π½
Proof of Theorem 4.1 π It is equivalent to prove that there exist π0 β₯ 1, π΄π β (π½π [π₯]π½ ) β². By Lemma 4.2, we obtain π = β π
π½π (π₯)π½ such as π, π β π½π [π₯]π½ and (ππ , ππ ) the reduced π½-fractionary expansion of π. π
π
1
By proposition 4.1, |π β ππ | < |π π
(π½
(πβ1) (π+1) π
2 π|
. According to Corollary 4.3 and Lemma 4.5, we have
(πππ β πππ )) inπ½π [π₯]π½ .So, we obtain (πβ1)(π+1)
π 1 πππ β πππ |π½| < |π½ |(πβ1)(π+1)/π | |< |π| π |ππ | which implies that deg(ππ ) β€ deg(π) + (π β 1)(π + 1), where π
deg(ππ ) = β deg(π΄π ) β€ deg(π) + (π β 1)(π + 1). π=1
βππ=1(deg(π΄π ) β (π β 1)) β€ deg(π) + (π β 1). Finally there exists π0 β₯ 1, such that, for Thus deg(π΄π ) β (π β 1) β€ 0, for all π β₯ π0 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 πΉπ ((π₯ β1 )), 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.
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