PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 100(114) (2016), 279–285 DOI: 10.2298/PIM1614279M

PURELY PERIODIC β-EXPANSIONS IN CUBIC SALEM BASE IN 𝔽_{q}((X^{−1} ))

Faïza Mahjoub

Abstract. Let 𝔽_{q} be the finite field with q elements and β Salem series in 𝔽_{q}((X^{−1} )). It is proved in [15 ] that, in this case, all elements in 𝔽_{q}(X, β) have purely periodic β-expansion. We characterize the formal power series f in 𝔽_{q}(X, β) with purely periodic β-expansions by the conjugate vector _{f} when β is a cubic unit. No similar results exist in the real case.

1. Introduction

Let β > 1 be a real number. The β-expansion of a real number x ∈ [0, 1] is defined as the sequence (x_{i})_{i⩾1} with values in {0, 1, ⋯, [β]} produced by the β-transformation T_{β} : x → βx (mod 1) as follows:

∀i ⩾ 1, x_{i} = [βT_{β}^{i−1} (x)], and thus x = ∑_{i⩾1} \frac{x_{i}}{β^{i}} .

This expansion was first introduced by Rényi [14]. A β-expansion is periodic if there exists p ⩾ 1 and m ⩾ 1 such that x_{k} = x_{k+p} , holds for all k ⩾ m. When x_{k} = x_{k+p} holds for all k ⩾ 1, then it is purely periodic. We denote by P er(β) the numbers in [0,1) with periodic β-expansions, Pur(β) the numbers in [0,1) with purely periodic β-expansions and Fin(β) the numbers in [0,1) with finite β-expansions.

Let ℚ(β) be the smallest field containing ℚ and β. An easy argument shows that Per(β) ⊆ ℚ(β) ∩ [0, 1) for every real number β > 1. In [17], Schmidt showed that if β is a Pisot number (an algebraic integer whose conjugates have modulus <1), then Per(β)=ℚ(β) ∩ [0, 1).

Ito and Rao discussed the purely periodic β-expansions in the statement [9] and they characterized all reals in [0, 1[ having purely periodic β-expansions with Pisot unit base. In [6], Berthé and Siegel completed the characterization in the Pisot non-unit base.

Set

γ(β) = sup{c ∈ [0, 1) : ∀r ∈ ℚ ∩ [0, c], d_{β}(r) is purely periodic}.

2010 Mathematics Subject Classification: 11R06; 37B50.

Key words and phrases: formal power series, finite fields, β-expansion, Salem element.

Communicated by Žarko Mijajlović.

Akiyama proved in [3] that if β is a Pisot unit number satisfying the finiteness property (Fin(β) = ℤ[β^{−1} ] ∩ ℝ_{+} ), then γ(β) > 0.

In the quadratic case, Schmidt [17] proved that if β satisfied β^{2} = nβ + 1 for some integer n ⩾ 1, then γ(β) = 1. Until now, it has been clear that for only known family is the γ(β) of which equals 1. Authors of [2] have proved that if β is not a Pisot unit, then γ(β) = 0. They showed that if β is a cubic Pisot unit satisfying the finiteness property such that the number field ℚ(β) is not totally real, then 0 < γ(β) < 1.

In 2006, Hbaib and Mkaour [8] introduced the β-expansion in the field of formal power series over a finite field 𝔽_{q}. They developed some results concerning the βexpansion of unity. Later, Scheicher [15] proved that P er(β) = 𝔽_{q}(X, β) if and only if β is a Pisot or Salem series. In [1], Abbes and Hbaib gave families of Pisot and Salem elements β in 𝔽_{q}((X^{−1} )) with the curious property that the β-expansion of any rational series in the unit disk D(0, 1) is purely periodic. Ghorbel, Hbaib and Zouari showed in [7] that if β is a quadratic Pisot unit base, then every rational f in the unit disk has a purely periodic β-expansion.

In [5], the authors proved that the β-expansion of any rational element in the unit disk D(0, 1) is purely periodic when β is a Pisot or Salem unit series in 𝔽_{q}((X^{−1} )).

Here, we continue in the same context: we take β a cubic Salem unit series in 𝔽_{q}((X^{−1} )) and characterize the formal power series f ∈ 𝔽_{q}(X, β) with purely periodic β-expansions by the norm of the conjugate vector f̃.

Our work is organized as follows: In section 2, we introduce the field of formal power series over a finite field 𝔽_{q} and the β-expansion theory in this field. In Section 3, we give our main result with its proof.

2. β-expansions in 𝔽_{q}((X^{−1} ))

Let 𝔽_{q} be the finite field with q elements, 𝔽_{q}[X] the ring of polynomials with coefficients in 𝔽_{q}, 𝔽_{q}(X) the field of rational functions, 𝔽_{q}(X, β) the minimal extension of 𝔽_{q} containing X and β and 𝔽_{q}[X, β] the minimal ring containing X and β. Let 𝔽_{q}((X^{−1} )) be the field of formal power series of the form:

f = ∑_{k=−∞}^{l} f_{k}X^{k}, f_{k} ∈ 𝔽_{q}

where

l = degf := \{ \begin{matrix}max\{k : f_{k} ≠ 0\} & for f ≠ 0; \\ −∞ & for f = 0. \end{matrix}

We define the absolute value by

|f| = \{ \begin{matrix}q^{deg f} & for f ≠ 0; \\ 0 & for f = 0. \end{matrix}

PURELY PERIODIC β-EXPANSIONS IN CUBIC SALEM BASE IN 𝔽_{q}((X^{−1} )) 281

Since |.| is not archimedean, |.| fulfills the strict triangle inequality

|f + g| ⩽ max(|f|, |g|) and
|f + g| = max(|f|, |g|) if |f| ≠ |g|.

Let f ∈ 𝔽_{q}((X^{−1} )), define the integer (polynomial) part [f] = ∑ _{k=0}^{l} f_{k}X^{k} where the empty sum, as usual, is defined to be zero. Therefore [f] ∈ 𝔽_{q}[X] and (f − [f]) is in the unit disk D(0, 1) for all f ∈ 𝔽_{q}((X^{−1} )).

Lemma 2.1. Let f = ∑ _{i⩾1} \frac{α_{i}}{X^{i}} ∈ 𝔽_{q}((X^{−1} ))∩D(0, 1). Then (α_{i})_{i⩾1} is periodic if and only if f ∈ 𝔽_{q}(X).

Lemma 2.2. Let f = ∑ _{i⩾1} \frac{α_{i}}{X^{i}} ∈ 𝔽_{q}((X^{−1} )) ∩ D(0, 1). Then (α_{i})_{i⩾1} is purely periodic if and only if f ∈ 𝔽_{q}(X) and 0 is not a pole of f.

Proposition 2.1. [13] Let K be complete field with respect to (a non archimedean absolute value |.|) and L/K (K ⊂ L) be an algebraic extension of degree m. Then |.| has a unique extension to L defined by : |a| = \sqrt[m]{|N_{L/K} (a)|} and L is complete with respect to this extension.

We apply Proposition 2.1 to algebraic extensions of 𝔽_{q}((X^{−1} )). Since 𝔽_{q}[X] ⊂ 𝔽_{q}((X^{−1} )), every algebraic element over 𝔽_{q}[X] can be evaluated. However, since 𝔽_{q}((X^{−1} )) is not algebraically closed and such an element is not necessarily expressed as a power series over X^{−1} . For a full characterization of the algebraic closure of 𝔽_{q}[X], we refer to Kedlaya [10].

An element β = β_{1} ∈ 𝔽_{q}((X^{−1} )) is called a Pisot (respectively, Salem) element if it is an algebraic integer over 𝔽_{q}[X], |β| > 1 and |β_{j} | < 1 holds for all its conjugates β_{j} (respectively, |β_{j} | ⩽ 1 and there exists at least one conjugate β_{k} such that |β_{k}| = 1).

Bateman and Duquette [4] characterized the Pisot and Salem elements in 𝔽_{q}((X^{−1} )):

Theorem 2.1. Let β ∈ 𝔽_{q}((X^{−1} )) be an algebraic integer over 𝔽_{q}[X] and
P (y) = y^{n} − A_{1}y^{n−1} − ⋯− A_{n}, A_{i} ∈ 𝔽_{q}[X],

be its minimal polynomial. Then (i) β is a Pisot element if and only if |A_{1}| > max_{2⩽i⩽n} |A_{i}|

(ii) β is a Salem element if and only if |A_{1}| = max_{2⩽i⩽n} |A_{i}|.

Let β, f ∈ 𝔽_{q}((X^{−1} )) with |β| > 1. A representation in base β (or βrepresentation) of f is an infinite sequence (d_{i})_{i⩾1}, d_{i} ∈ 𝔽_{q}[X], such that

f = ∑_{i⩾1} \frac{d_{i}}{β^{i}} .

A particular β-representation of f is called the β-expansion of f in base β, noted d_{β}(f), which is obtained by using the β-transformation T_{β} in the unit disk which is given by T_{β}(f) = βf − [βf]. Then d_{β}(f) = (a_{i})_{i⩾1} where a_{i} = [βT_{β}^{i−1} (f)].

An equivalent definition of the β-expansion can be obtained by a greedy algorithm. This algorithm works as follows. We set r_{0} = f, a_{i} = [βr_{i−1} ] and r_{i} = βr_{i−1} −a_{i} for all i ⩾ 1. The β-expansion of f will be noted as d_{β}(f) = (a_{i})_{i⩾1}.

We note that d_{β}(f) is finite if and only if there is a k ⩾ 0 such that T^{k}(f) = 0, d_{β}(f) is ultimately periodic if and only if there is some smallest p ⩾ 0 (the preperiod length) and s ⩾ 1 (the period length) for which T_{β}^{p+s} (f) = T_{β}^{p}(f).

Now, let f ∈ 𝔽_{q}((X^{−1} )) be an element with |f| ⩾ 1. Then there is a unique k ∈ ℕ such that |β|^{k} ⩽ |f| < |β|^{k+1} . Hence |\frac{f}{β^{k+1}} | < 1 and we can represent f by shifting d_{β}(\frac{f}{β^{k+1}} ) by k digits to the left. Therefore, if d_{β}(f) = 0.d_{1}d_{2}d_{3} …, we obtain d_{β}(βf) = d_{1}.d_{2}d_{3} . . .. If we have d_{β}(f) = d_{l}d_{l−1} …d_{0}.d_{−1} …d_{m}, then we put deg_{β}(f) = l and ord_{β}(f) = m. In the sequel, we will use the following notations:

• Fin(β) = {f ∈ 𝔽_{q}((X^{−1} )) : d_{β}(f) is finite}.

• Per(β) = {f ∈ 𝔽_{q}((X^{−1} )) : d_{β}(f) is eventually periodic}.

• Pur(β) = {f ∈ 𝔽_{q}((X^{−1} )) and |f| < 1 : d_{β}(f) is purely periodic}.

Remark 2.1. In contrast to the real case, there is no carry occurring, when we add two digits. Therefore, if z, w ∈ 𝔽_{q}((X^{−1} )), we have d_{β}(z + w) = d_{β}(z) + d_{β}(w) digitwise.

Theorem 2.2. [8] A β-representation (d_{j})_{j⩾1} is the β-expansion of f in the unit disk if and only if |d_{j} | < |β| for j ⩾ 1.

In the field of formal series case, Scheicher, Jellali and Mkaouar [16] had studied the characterization of purely periodic β-expansions in the Pisot unit base. Later, Hbaib-Mkaouar and Scheicher proved independently the following:

Theorem 2.3. [15] β is a Pisot or Salem element if and only if Per(β) = 𝔽_{q}(X, β).

Theorem 2.4. [8] β is Pisot or Salem element if and only if d_{β}(1) is periodic.

The authors of [5] gave the following result.

Theorem 2.5. [5] Let β be a Pisot or Salem unit series in 𝔽_{q}((X^{−1} )) and r ∈ 𝔽_{q}(X) ∩ D(0, 1). Then d_{β}(r) is purely periodic.

In [11] and [12], metric results were established and the relation to continued fractions was studied. Stating

γ(β) = sup{c ∈ [0, 1) : ∀f ∈ 𝔽_{q}(X) ∩ D(0, c), d_{β}(f) is purely periodic}.

The study of the quality γ(β) in 𝔽_{q}((X^{−1} )) was interesting for some researchers in the last years. Specifically, we have the following theorems.

Theorem 2.6. [1] Let β be a Pisot or Salem unit series. Then γ(β) > 0.

Theorem 2.7. [7] If β is a quadratic Pisot unit series, then γ(β) = 1

3. Results

Let β be an algebraic unit series of minimal polynomial β^{d}+A_{d−1} β^{d−1} +⋯ +A_{0} where A_{i} ∈ 𝔽_{q}[X] for i ∈ { 1, …, d − 1} and A_{0} ∈ 𝔽_{q}^{∗}. Let β_{2}, …, β_{d} be the conjugates of β. For f = r_{0} + r_{1}β + ⋯+ r_{d−1} β^{d−1} ∈ 𝔽_{q}(X, β), we define f_{i} = r_{0} + r_{1}β_{i} + ⋯+ r_{d−1} β_{i}^{d−1} ∈ 𝔽_{q}(X, β) with 2 ⩽ i ⩽ d and f̃ the conjugate vector of

f by f̃ = ( ⋮_{f_{d}}^{f_{2}} ) and ∥f̃∥ = sup_{2⩽k⩽d} |f_{k}|.

Theorem 3.1. Let β be a Pisot unit series and f ∈ 𝔽_{q}(X, β) ∩ D(0, 1). If d_{β}(f) is purely periodic, then ∥f̃∥ < |β|.

Proof. We have d_{β}(f) = \overline{a_{1}, ⋯, a_{s}} with a_{i} ∈ 𝔽_{q}[X]. Then we can write

f = \frac{a_{1}}{β} + ⋯+ \frac{a_{s}}{β^{s}} + \frac{1}{β^{s}} (f).

Hence,

f(1 − \frac{1}{β^{s}} ) = \frac{a_{1}}{β} + ⋯+ \frac{a_{s}}{β^{s}} .

Therefore, we obtain

(β^{s} − 1)f = a_{1}β^{s−1} + ⋯+ a_{s}.

Let (β_{j})_{2⩽j⩽d} the conjugates of β. Afterward |β_{j} | < 1 for all 2 ⩽ j ⩽ d which leads to

(β_{j}^{s} − 1)f_{j} = a_{1}β_{j}^{s−1} + ⋯+ a_{s}.

Then, we get

|f_{j} | = |a_{1}β_{j}^{s−1} + ⋯+ a_{s}| < |β|.

Finally, ∥f̃∥ < |β|. □

Remark 3.1. The same arguments as in the proof of the last theorem, one can prove that if β is a Salem unit series and f ∈ 𝔽_{q}(X, β) ∩ D(0, 1), and if d_{β}(f) is purely periodic then ∥f̃∥ ⩽ |β|.

Now, here is our main theorem.

Theorem 3.2. Let β be a cubic Salem unit series in 𝔽_{q}((X^{−1} )) and f ∈ 𝔽_{q}(X, β) ∩ D(0, 1). ∥f̃∥ < |β| if and only if d_{β}(f) is purely periodic.

Proof. The sufficient condition is deduced by Theorem 3.1. As for the necessary condition, let f = r_{0} + r_{1}β + r_{2}β^{2} ∈ 𝔽_{q}(X, β) ∩ D(0, 1) and f_{1} , f_{2} the two conjugates of f in 𝔽_{q}(X, β). Let f_{i} = r_{0} + r_{1}β_{i} + r_{2}β_{i}^{2} such that |f_{i}| = ∥f̃∥. Since, we have |r_{0} + r_{1}β + r_{2}β^{2}| < 1. Then we obtain two cases:

Case 1: |r_{0}| < 1 and |r_{1}β + r_{2}β^{2}| < 1. If |r_{0}| < 1 then thanks to Theorem 2.5 d_{β}(r_{0}) is purely periodic. If |r_{1}β + r_{2}β^{2}| < 1 then |r_{1} + r_{2}β| < \frac{1}{|β|} < 1. Hence, we obtain two subcases:

(i) |r_{1}| < 1 and |r_{2}β| < 1. If |r_{1}| < 1 then by Theorem 2.5 d_{β}(r_{1}) is purely periodic. So, d_{β}(r_{1}β) is purely periodic. If |r_{2}β| < 1 then |r_{2}| < \frac{1}{|β|} < 1. Therefore, from Theorem 2.5 d_{β}(r_{2}) is purely periodic, consequently d_{β}(r_{2}β^{2}) is purely periodic.

(ii) |r_{1}| > 1, |r_{2}β| > 1 and |r_{1}| = |r_{2}β|. If |r_{1}| = |r_{2}β| then |r_{1}β_{i}| = |r_{2}ββ_{i}| > |r_{2}β_{i}^{2}|. Reminding that we have |r_{0} + r_{1}β_{i} + r_{2}β_{i}^{2}| < |β| and |r_{0}| < 1 < β, we obtain |r_{1}β_{i} + r_{2}β_{i}^{2}| < |β|. However, when |β_{i}| = 1 we get |r_{1}β_{i} + r_{2}β_{i}^{2}| = |r_{1}β_{i}| = |r_{1}|. Thus, |r_{1}| < |β| and d_{β}(r_{1}) is purely periodic and it follows that d_{β}(r_{1}β) is purely periodic. Moreover, we have |r_{1}| = |r_{2}β| then |r_{2}| < 1. Using Theorem 2.5 we get d_{β}(r_{2}) is purely periodic. Then d_{β}(r_{2}β^{2}) is purely periodic. Case 2: |r_{0}| > 1, |r_{1}β + r_{2}β^{2}| > 1 and |r_{0}| = |r_{1}β + r_{2}β^{2}|. Reminding that

we have |r_{1}β + r_{2}β^{2}| > |r_{1}β_{i} + r_{2}β_{i}^{2}|, we get |r_{0}| < |β|. Therefore d_{β}(r_{0}) is purely periodic. We have |r_{1}β + r_{2}β^{2}| = |r_{0}| < |β| thereby |r_{1} + r_{2}β| < 1. Thus we obtain two subcases.

(i) |r_{1}| < 1 and |r_{2}β| < 1. If |r_{1}| < 1 then from Theorem 2.5 d_{β}(r_{1}) is purely periodic. Consequently d_{β}(r_{1}β) is purely periodic. If |r_{2}β| < 1 then |r_{2}| < \frac{1}{|β|} < 1. Afterward, we use Theorem 2.5 which asserts that d_{β}(r_{2}) is purely periodic, moreover d_{β}(r_{2}β^{2}) is purely periodic.

(ii) |r_{1}| > 1, |r_{2}β| > 1 and |r_{1}| = |r_{2}β|. We have |r_{0} + r_{1}β_{i} + r_{2}β_{i}^{2}| < |β| and |r_{0}| < |β|, we get |r_{1}β_{i} + r_{2}β_{i}^{2}| < |β| afterward |r_{1} + r_{2}β_{i}| < |β|. Then we obtain two subsubcases.

– |r_{1}| < |β| and |r_{2}β_{i}| < |β|. If |r_{1}| < |β| then d_{β}(r_{1}) is purely periodic, consequently d_{β}(r_{1}β) is purely periodic. If |r_{2}β_{i}| < |β| then |r_{2}| < |β| when |β_{i}| = 1, we get d_{β}(r_{2}) is purely periodic and it follows that d_{β}(r_{2}β^{2}) is purely periodic.

– |r_{1}| > |β|, |r_{2}β_{i}| > |β| and |r_{1}| = |r_{2}β_{i}| we obtain |r_{1}| = |r_{2}| when |β_{i}| = 1.

However, |r_{1}| = |r_{2}β| afterward |β| = 1 impossible. Since, we obtained in the last cases that d_{β}(r_{0}), d_{β}(r_{1}β) and d_{β}(r_{2}β^{2}) are purely

periodic, one can deduce that the desired result ( d_{β}(f) is purely periodic). □

Acknowledgments. I would like to thank Prof. M. Hbaib for reading the manuscript and his helpful discussions. I am also grateful to the referee for his/her comments.

References

1. F. Abbes, M. Hbaib, Rational Laurent series with purely periodic β-expansions, Osaka J. Math. 50 (2013), 807–816.

2. B. Adamczewski, C. Frougny, A. Siegel and W. Steiner, Rational numbers with purely periodic beta-expansion, Bull. Lond. Math. Soc. 42 (2010), 538–552.

3. S. Akiyama, Pisot number and greedy algorithm, in: Number theory, Diophantine, Computational and Algebraic Aspects, de Gruyter, Berlin, (1998), 9–21.

4. P. Bateman, A. L. Duquette, The analogue of Pisot–Vijayaraghvan numbers in fields of power series, Ill. J. Math. 6 (1962) 594–606.

5. S. Ben Hariz, M. Hbaib, F. Mahjoub, Purely periodic β-expansions with Pisot or Salem unit base in 𝔽_{q}((X^{−1} )), Math. Z. DOI: 10.1007/s00209-016-1617-x.

6. V. Berthé, A. Siegel, Purely periodic β-expansions in the Pisot non unit case, J. Number Theory. 127 (2007), 153–172.

7. R. Ghorbel, M. Hbaib, S. Souari, Purly periodic beta-expansions over Laurent series, Int. J. of Algebra and Comput. 22 (2012), 1–12.

8. M. Hbaib, M. Mkaouar, Sur le bêta-développement de 1 dans le corps des séries formelles, Int. J. Number Theory. 2 (2006), 365–377.

9. S. Ito, H. Rao, Purely periodic β-expansions with Pisot unit base, Proc. Amer. Math. Soc. 133 (2005), 953–964.

10. K. S. Kedlaya, The algebraic closure of the power series field in positive characteristic, Proc. Amer. Math. Soc. 12 (2001), 3461–3470.

11. B. Li, J. Wu, Beta-expansions and cotinued fraction expansion over formal Laurent series, Finite Fields Appl. 14 (2008), 635–647.

12. B. Li, J. Wu , J. Xu, Metric properties and exceptional sets of β-expansions over formal Laurent series, Monatsh. Math. 155 (2008), 145–160.

13. J. Neukirch, Algebraic Number Theory, the Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin, 322 (1999).

14. A. Rènyi, Representations for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hung. 8 (1957),477–493.

15. K. Scheicher, Beta-expansions in algebraic function fields over finite fields, Finite Fields Appl. 13 (2007), 394–410.

16. K. Scheicher, M. Jellali, M. Mkaouar, Purely periodic β-expansions with Pisot Unit Base over Laurent Series, Int. J. Contemp. Math. Sciences. 3 (2008), 357–369.

17. K. Schmidt, On periodic expansions of Pisot numbers and Salem numbers, Bull. London Math. Soc. 12 (1980), 269–278.

Department of Mathematics (Received 20 03 2015) Higher institute of applied science and technology (Revised 08 12 2015) University of Monastir Mahdia Tunisia

 faiza.mahjoub@yahoo.fr