PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 98(112) (2015), 237–242 DOI: 10.2298/PIM141206012A

A TRANSCENDENCE CRITERION FOR CONTINUED FRACTION EXPANSIONS IN POSITIVE CHARACTERISTIC

Basma Ammous, Sana Driss, and Mohamed Hbaib

Abstract. We exhibit a family of transcendental continued fractions of formal power series over a finite field through some specific irregularities of its partial quotients.

1. Introduction

A well-known open question in diophantine approximation suggested by Khintchine in [5] asks whether an irrational algebraic number x of degree > 2 has a continued fraction expansion whose sequence of partial quotients is unbounded. The answer to this conjecture remains a hard matter. Several transcendence criteria for continued fractions that have been established recently gave a partial solution to this question. In [2] Baker proved that if x = [a_{0}, a_{1}, a_{2}, …] such that a_{n} = a_{n+1} = ⋯= a_{n+λ(n)−1} , for infinitely many positive integers n where λ(n) is a sequence of integers verifying certain increasing properties, then x is transcendental. The proof of this result is based on Liouville’s and Roth’s theorems.

Recently and based on the Schmidt Subspace Theorem, Adamczewski and Bugeaud [1] improved the result of Baker.

In 1967, Schmidt [12] demonstrated that any positive irrational number which is very well approximated by quadratic numbers is either quadratic or transcendental. This result has been used in several works.

However, for formal power series over a finite field, we have some examples of algebraic formal series of degree ⩾ 3 whose sequence of the degrees of the partial quotients is bounded, as well as examples whose partial quotients take an infinity of values.

In 1976, Baum and Sweet [3] gave the first example of algebraic formal series of degree 3 in 𝔽_{2}((X^{−1} )) whose partial quotients have only a finite number of values. This work was pursued in [7] by Mills and Robbins who provided an example of algebraic formal series over 𝔽_{2}((X^{−1} )) whose sequence of partial quotients is unbounded. Moreover, Robbins gave a family of cubic formal power series with bounded partial quotients [11].

2010 Mathematics Subject Classification: 11A55, 11J81.

Key words and phrases: continued fraction; formal power series; transcendence.

Communicated by Žarko Mijajlović.

In 2004, Mkaouar [9] gave a new transcendence criteria of formal power series over a finite field that is based on the degree of its partial quotients.

Theorem 1.1. [9] Let f ∈ 𝔽_{q}((X^{−1} )) be an irrational formal series which is not quadratic such that

f = [\overbrace{a_{1}, …, a_{1}}^{n_{1}}, \overbrace{a_{2}, …, a_{2}}^{n_{2}}, \overbrace{a_{3}, …, a_{3}}^{n_{3}}, …],

where a_{i} are blocks of consecutive partial quotients. Let r_{i} be the sum of degrees of partial quotients of block a_{i}. If

lim_{∞} \frac{n_{i}r_{i}}{n_{i−1} r_{i−1}} = lim supn_{i} = +∞,

then f is transcendental.

In [4], Hbaib, Mkaouar and Tounsi constructed a family of transcendental continued fractions over 𝔽_{q}((X^{−1} )) from an algebraic formal power series of degree more than 2.

Theorem 1.2. [4] Let g be an algebraic formal power series such that deg(g)>0 and f = [B_{1}, B_{2}, …] where B_{i} are finite blocks of partial quotients whose the first n_{i}-terms are those of the continued fraction expansion of g. Let d_{i} denote the sum of degrees of B_{i} and δ_{i} the sum of degrees of the first n_{i}-terms of B_{i}. If

lim inf_{s→+∞} \frac{1}{δ_{s}} ∑_{j=1}^{s−1} d_{j} = 0,

then f is transcendental or quadratic.

Our main purpose here is twofold: to improve the last results and to give a new transcendence criteria depending only on the length of specific blocks appearing in the sequence of partial quotients. The present paper is organized as follows: in Section 2, we define the field of formal series and the continued fraction expansions over this field. In Section 3, we state the main transcendence criterion and we present some lemmas that we will use to prove our result. We close this section with the proof of our main theorem (see Theorem 3.1) and an example to illustrate the limit of our result.

2. Field of formal series 𝔽_{q}((X^{−1} ))

Let 𝔽_{q} be a field with q > 1 elements of characteristic p > 0, 𝔽_{q}[X] the ring of polynomials with coefficient in 𝔽_{q} and 𝔽_{q}(X) the field of rational functions. Let 𝔽_{q}((X^{−1} )) = {f = ∑ _{n⩾n_{0}} b_{n}X^{−n} | b_{n} ∈ 𝔽_{q}, n_{0} ∈ ℤ} be the field of formal power series. Define the absolute value

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

A TRANSCENDENCE CRITERION FOR CONTINUED FRACTION EXPANSIONS… 239

Thus, | · | is not an archimedean absolute value over 𝔽_{q}((X^{−1} )), that is |f + g| ⩽ max(|f|, |g|) and |f + g| = max(|f|, |g|) if |f| ≠ |g|. By analogy with the real case, we have a chain-fraction algorithm in 𝔽_{q}((X^{−1} )). A formal power series f = ∑ _{n⩾n_{0}} b_{n}X^{−n} has a unique decomposition as f = [f] + {f} with [f] ∈ 𝔽_{q}[X] and |{f}| < 1. The polynomial [f] is called the polynomial part of f and {f} is called the fractional part of f. We can write for any f ∈ 𝔽_{q}((X^{−1} ))

f = a_{0} + \frac{1}{a_{1} + \frac{1}{a_{2} + \frac{1}{...}}} = [a_{0}, a_{1}, a_{2}, …],

where a_{0} = [f] and a_{i} = [f_{i}] ∈ 𝔽_{q}[X] with deg(a_{i}) ⩾ 1 for any i ⩾ 1 and f_{i} = 1/{f_{i−1} }. The sequence (a_{i})_{i⩾0} is called the partial quotients of f and we denote by f_{n} = [a_{n}, a_{n+1} , …] the n-th complete quotient of f.

Remarks. 1) If (deg(a_{i}))_{i⩾0} is bounded, then f is said to have a bounded continued fraction expansion.

2) The expansion is finite if and only if f ∈ 𝔽_{q}(X).

3) The sequence of partial quotients of f is ultimately periodic if and only if f is quadratic over 𝔽_{q}(X).

Now, we define two sequences of polynomials (P_{n})_{n⩾0} and (Q_{n})_{n⩾0} by
P_{0} = a_{0}, Q_{0} = 1, P_{1} = a_{0}a_{1} + 1, Q_{1} = a_{1}
P_{n} = a_{n}P_{n−1} + P_{n−2} , Q_{n} = a_{n}Q_{n−1} + Q_{n−2} , for n ⩾ 2.

We easily check that

P_{n}Q_{n−1} − P_{n−1} Q_{n} = (−1)^{n−1} , for n ⩾ 1,

\frac{P_{n}}{Q_{n}} = [a_{0}, a_{1}, a_{2}, …, a_{n}], for n ⩾ 0.

P_{n}/Q_{n} is called the n^{th}convergent of f and it satisfies

lim_{n→∞} \frac{P_{n}}{Q_{n}} = f = [a_{0}, a_{1}, …, a_{n}, …].

With the nonarchimedean absolute value, we find the following important equality

|f − \frac{P_{n}}{Q_{n}} | = | \frac{P_{n+1}}{Q_{n+1}} − \frac{P_{n}}{Q_{n}} | = |Q_{n}Q_{n+1} |^{−1} = |a_{n+1} |^{−1} |Q_{n}|^{−2} .

Let f be an algebraic formal power series of minimal polynomial P (Y) = A_{m}Y^{m} + A_{m−1} Y^{m−1} + ⋯+ A_{0} where A_{i} ∈ 𝔽_{q}[X]. Set H(f) = max_{0⩽i⩽m} |A_{i}| and σ(f) = A_{m}.

Recall from [8] that a polynomial P ∈ 𝔽_{q}[X][Y] is said to be reduced if deg(A_{m−1} ) > deg(A_{i}) for any i ≠ m − 1, and an algebraic formal power series is reduced if its minimal polynomial is reduced and [f] ≠ 0.

In [4], the authors gave the following lemma which identifies the reduced formal power series.

Lemma 2.1. Let f be an algebraic formal power series of degree d and P its minimal polynomial. We denote by f_{1}, …, f_{d−1} the conjugates of f in the algebraic closure of 𝔽_{q}((X^{−1} )). Then f is reduced if and only if |f| > 1 and |f_{i}| < 1, for all i ∈ {1, …, d − 1}.

3. Results

Before giving the main result, we need to introduce some notation. If K_{n} = u_{α_{0}} u_{α_{2}} …u_{α_{n}} is a finite block formed by n + 1 polynomials, we denote by |K_{n}| its length and by φ(K_{n}) the maximal degree which appears in the terms of K_{n}, which means that φ(K_{n}) = max_{0⩽i⩽n} (deg(u_{α_{i}} )). If U_{n}, V_{n} are two finite blocks of polynomials, we write U_{n}V_{n} for the block resulting by concatenation of them.

Theorem 3.1. Let f ∈ 𝔽_{q}((X^{−1} )) such that f = [U_{0}V_{0}U_{1}V_{1} …U_{n}V_{n} …] where (U_{n})_{n⩾0} and (V_{n})_{n⩾0} are two sequences of finite blocks of polynomials such that

1) U_{i} = P_{i}P_{i}^{q}P_{i}^{q^{2}} …P_{i}^{q^{λ_{i} −1}} , for any i ⩾ 0, with P_{i} ∈ 𝔽_{q}[X] of degree ⩾ 1.

2) The sequence (|V_{n}|/|U_{n}|)_{n⩾0} is bounded.

3) (λ_{i})_{i⩾0} is an increasing sequence of positive integers.

4) (deg(P_{i}))_{i⩾0} is bounded.

5) φ(V_{n}) ⩽ φ(U_{n}), for all n ⩾ 0.

If f satisfies

lim sup_{n→∞} \frac{q^{λ_{n} −λ_{n−1}}}{nλ_{n−1}} = +∞,

then f is transcendental.

The proof of this theorem breaks into four lemmas.

Lemma 3.1. [4] Let f be an algebraic formal power series of degree d such that f = [a_{1}, a_{2}, …, a_{t}, h] where a_{1}, …, a_{t} ∈ 𝔽_{q}[X], h ∈ 𝔽_{q}((X^{−1} )). If |f| ⩾ 1 and |h| > 1, then h is algebraic of degree d and

H(h) ⩽ H(f)|∏_{i=1}^{t} a_{i} |^{d−2} .

Lemma 3.2. Let P (Y) = A_{n}Y^{n} +A_{n−1} Y^{n−1} +⋯ +A_{0} be a reduced polynomial with A_{i} ∈ 𝔽_{q}[X]. If A_{0} ≠ 0, then P is irreducible.

Proof. Let f be the unique root of P such that |f| > 1 and assume that P (Y) is reducible; then P (Y) = P_{1}(Y)P_{2}(Y) with P_{1}, P_{2} ∈ 𝔽_{q}[X][Y]. We suppose that P_{1}(f) = 0; then from Lemma 2.1, all the roots of P_{2} have absolute values < 1, so the constant coefficient in P_{2} is equal to 0, which is absurd because 0 is not a root of P. □

Lemma 3.3. [10] Let f = [a_{0}, a_{1}, …] and g = [b_{0}, b_{1}, …] be two formal series having the same first n + 1 partial quotients. Then

|f − g| ⩽ \frac{1}{|Q_{n}|^{2}} .

Lemma 3.4. [4] Let f and g be two algebraic formal power series of degrees d and m respectively. If g is reduced and f ≠ g, then

|f − g| ⩾ \frac{1}{H(f)^{m}|g|^{d−2} |σ(g)|^{max (m−1,m(d−m+2)−1)}} .

Proof. Assume contrary that f is algebraic of degree d > 2. Let us use the notation: λ_{n} = |U_{n}|, s_{n} = |V_{n}|, for all n ⩾ 0 and α_{n} = ∑ _{i=0}^{n−1} (λ_{i} +s_{i}), for all n ⩾ 1. Let g_{n} denote the continued fraction [P_{n}, P_{n}^{q}, P_{n}^{q^{2}}, P_{n}^{q^{3}}, …]. An easy calculation ensures that g_{n} verifies the following equation

g_{n}^{q+1} − P_{n}g_{n}^{q} − 1 = 0.

Hence Lemma 3.2 guarantees that g_{n} is algebraic of degree q + 1. Let f_{α_{n}} = [U_{n}V_{n}U_{n+1} V_{n+1} …] denote the α_{n}^{th} complete quotient of f. Since (deg(P_{i}))_{i⩾0} is bounded, then for sufficiently large n, g_{n} ≠ f_{α_{n}} . On the other hand, it follows from Lemma 3.1 that f_{α_{n}} is algebraic of degree d > 2. Therefore, according to Lemma 3.4, we infer that

|f_{α_{n}} − g_{n}| ⩾ H(f_{α_{n}} )^{−q−1} |g_{n}|^{2−d} .

Moreover, by using again Lemma 3.1, we can check, for sufficiently large n that

(3.1)  |f_{α_{n}} − g_{n}| ⩾ H(f)^{−q−1} |P_{n}|^{2−d} | ∏_{i=0}^{α_{n} −1} a_{i} |^{(d−2)(−q−1)} ,

where (a_{i})_{i⩾0} is the sequence of partial quotients of f.

Furthermore, f_{α_{n}} and g_{n} have the same first λ_{n} partial quotients, hence Lemma 3.3 implies that

(3.2)  |f_{α_{n}} − g_{n}| ⩽ |P_{n}P_{n}^{q}P_{n}^{q^{2}} …P_{n}^{q^{λn −1}} |^{−2} .

Combining (3.1) and (3.2), we get

|P_{n}P_{n}^{q}P_{n}^{q^{2}} …P_{n}^{q^{λn −1}} |^{2} ⩽ H(f)^{q+1} |P_{n}|^{d−2} | ∏_{i=0}^{α_{n} −1} a_{i} |^{(d−2)(q+1)}

whence

2 deg(P_{n})( \frac{q^{λ_{n}} − 1}{q − 1} ) ⩽ (q+1) log_{q} H(f)+(q+1)(d−2) ∑_{i=0}^{α_{n} −1} deg(a_{i})+(d−2) deg(P_{n}).

The fact that (deg(P_{i}))_{i⩾0} is bounded yields the inequality

lim sup_{n→∞} \frac{q^{λ_{n}}}{∑ _{i=0}^{α_{n} −1} deg(a_{i})} ⩽ C, with C = (q^{2} − 1)(d − 2).

Set h = sup_{i⩾0}(deg(P_{i})). As φ(V_{i}) ⩽ φ(U_{i}) for all i ⩾ 0, we get deg(a_{i}) ⩽ q^{λ_{n−1} −1} h, for all 0 ⩽ i ⩽ α_{n}. Therefore

lim sup_{n→∞} \frac{q^{λ_{n}}}{q^{λ_{n−1} −1} hα_{n}} ⩽ C.

Since the sequence (|V_{i}|/|U_{i}|)_{i⩾0} is bounded, there exists c > 0 such that s_{i} < cλ_{i} for all i ⩾ 0. Thus, α_{n} < (c + 1)nλ_{n−1} . Hence, we conclude that

lim sup_{n→∞} \frac{q^{λ_{n} −λ_{n−1}}}{nλ_{n−1}} < ∞, the desired contradiction.
□

We close the paper with the following example.

Example. Let f ∈ 𝔽_{2}((X^{−1} )) such that f = [U_{0}V_{0}U_{1}V_{1} …U_{n}V_{n} …] where U_{i} = [P_{i}, P_{i}^{2}, P_{i}^{4}, …, P_{i}^{2^{λ_{i} −1}} ], with P_{i} = X + i and V_{i} = [X, X^{2}, X, X^{2}, …, X, X^{2}] of length λ_{i} = (i + 1)^{2}, for all i ⩾ 0. Then f is transcendental because

lim sup_{n→∞} \frac{2^{2n+1}}{n^{3}} = +∞.

Acknowledgements. The authors would like to thank the referee for many useful comments. We also thank Prof N. Jarboui for his valuable remarks.

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Department of Mathematics (Received 30 01 2014) Faculty of Sciences (Revised 20 08 2014 and 11 11 2014) Sfax Tunisia

 ammous.basma@hotmail.fr
 sana_driss@yahoo.fr
 mmmhbaib@gmail.com