Annales de l’institut Fourier (AIF)

Peláez, José Ángel

Rättyä, Jouni

Weighted norm inequalities for derivatives on Bergman spaces

Chiba, Hayato

Weights, Kovalevskaya exponents and the Painlevé property

Hu, Yining

Lasjaunias, Alain

Université Grenoble Alpes
ANNALES DE
L’INSTITUT FOURIER
Yining Hu & Alain Lasjaunias
Period-doubling Continued Fractions are Algebraic in
Characteristic 2
Tome 74, n
o
5 (2024), p. 1809-1817.
https://doi.org/10.5802/aif.3614
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C EN T R E
MER S ENN E
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Ann. Inst. Fourier, Grenoble
74, 5 (2024) 1809-1817
PERIOD-DOUBLING CONTINUED FRACTIONS ARE
ALGEBRAIC IN CHARACTERISTIC 2
by Yining HU & Alain LASJAUNIAS (*)
Abstract. — For an arbitrary pair of distinct and non constant polynomials, a
and b in F2[t], we build a continued fraction in F2((1/t)) whose partial quotients are
only equal to a or b. In a previous work of the first author and Han, the authors
considered two cases where the sequence of partial quotients represents in each
case a famous and basic 2-automatic sequence, both defined in a similar way by
morphisms. They could prove the algebraicity of the corresponding continued frac-
tions for several pairs (a, b) in the first case (the Prouhet–Thue–Morse sequence)
and gave the proof for a particular pair for the second case (the period-doubling
sequence). Recently Bugeaud and Han proved the algebraicity for an arbitrary pair
in the first case. Here we give a short proof for an arbitrary pair in the second case.
Résumé. — Pour une paire de polynômes non constants et distincts a et b dans
F2[t], on construit une fraction continue dans F2((1/t)) dont les quotients partiels
sont égales à a ou b. Dans un travail précédent de la première auteure et Han,
les auteurs ont considéré deux cas où les quotients partiels forment des suites 2-
automatiques bien connues, définies de façon similaire comme point fixe des mor-
phismes. Ils ont pu démontré l’algébricité des fractions continues associées pour
plusieures paires (a, b) dans le premier cas (la suite de Prouhet–Thue–Morse) et
ont donné la preuve pour une paire dans le deuxième cas (la suite de doublement de
période). Récemment, Bugeaud et Han ont démontré l’algébricité pour une paire
arbitraire dans le premier cas. Ici, on donne une preuve courte pour une paire
arbitraire dans le deuxième cas.
1. Introduction
Let F2((1/t)) be the field of power series in 1/t, where t is a formal
indeterminate, over the finite field F2. A non-zero element is α = tn +∑
i<n ai · ti where n belongs to Z and ai = 0 or 1. An absolute value on
Keywords: Algebraicity, Automatic sequence, Continued fraction, Period-doubling
sequence.
2020 Mathematics Subject Classification: 11B85, 11J70, 11B50, 11Y65, 05A15, 11T55.
(*) This work is supported by the National Science Foundation of China grant 12001216.
1810 Yining HU & Alain LASJAUNIAS
F2((1/t)) is defined by |0| = 0 and |α| = |t|n where |t| > 1 is a fixed given
real number .
So F2((1/t)) is the completion of F2(t) for this absolute value. Every ir-
rational (resp. rational) element in F2((1/t)) can be expanded in an infinite
(resp. finite) continued fraction: α = [a0; a1, . . . , an, . . .] where the ai are in
F2[t] and deg(ai) > 0 for i > 0. For a basic introduction on fields of power
series and continued fractions, the reader may consult [6].
The origin of the question discussed here is due to G.-N. Han and the
first author. In [4], the authors considered two basic sequences, formed
by an infinite word with two letters a and b, belonging to the family of
2-automatic sequences (see [1, p. 173 and p. 176]) . Both sequences are
obtained in a similar way, as fixed point of a morphism. For the first one,
the (a; b)-Prouhet–Thue–Morse sequence, denoted by t, the morphism τ is
defined by τ(a) = ab and τ(b) = ba, and we have
t = τ∞(a) = abbabaabbaab . . .
Note that this famous sequence was considered very long ago and has
been the starting point of various studies. For the second one, the (a; b)-
period-doubling sequence, denoted by p, the morphism σ is defined by
σ(a) = ab and σ(b) = aa, and we have
p = σ∞(a) = abaaabababaaabaa . . .
In both cases, we consider a pair (a, b) of distinct and non-constant poly-
nomials in F2[t], and we can associate with each sequence an infinite contin-
ued fraction in F2((1/t)), CF(t) and CF(p), where the sequence of partial
quotients is derived from the sequences t and p :
αt = CF(t) = [a; b, b, a, b, a, a, . . .]
and
αp = CF(p) = [a; b, a, a, a, b, a, . . .].
It was proved in [4] that, for pairs (a, b) with deg a + deg b ⩽ 7, αt was
a root of a polynomial of degree 4, with five coefficients in F2[t] depending
on (a, b). In a recent work, extending the case of CF(t), Bugeaud and
Han [3] obtained the same result for all pairs (a, b) and they gave the
explicit formulas for the 5 coefficients as polynomials in a and b.
Also in [4], it was proved that for (a, b) = (t3, t2 + t + 1) we have
(t5 + t3 + t2) · α4p + (t8 + t6 + t5 + t3) · α3p + (t5 + t4 + t3) · α2p + 1 = 0.
ANNALES DE L’INSTITUT FOURIER
PERIOD-DOUBLING CONTINUED FRACTIONS ARE ALGEBRAIC 1811
In this note, we give a short proof of the general case for the sequence p
with the following theorem, thus confirming [4, Conjecture 1.5].
Theorem 1.1. — Let a, b be two distinct non constant elements in F2[t].
Let αp = [σ∞(a)] ∈ F2((1/t)). Define P (x) ∈ F2(t)[x] to be
(1.1) P (x) = Ax4 + Bx3 + Cx2 + 1
with
A = ab + b2 + 1, B = ab(a + b), C = ab.
Then P (αp) = 0.
Remark 1.2. — An elementary proof shows that P (x) has no solution in
F2(t). Consequently, since αp is not quadratic, P (x) is irreducible and αp
has degree 4 over F2(t).
Remark 1.3. — The theorem remains true when we replace F2 by any
other field K of characteristic 2.
Remark 1.4. — In fact we have proven that the continued fraction αp as
a series in the ring F2((1/a, 1/b)) is algebraic over F2(a, b).
In Section 2 we recall notation and formulas for continued fractions. In
Section 3 we give the proof of Theorem 1.1 and in a last section we make
some comments about the link with Riccati differential equations.
2. Notation and basic formulas for continued fractions
We use the same notation as in [6], which we recall in this section.
Let W = w1, w2, . . . , wn be a sequence of variables over a ring A. We set
|W | = n for the length of the word W . We define the following operators
for the word W .
W ′ = w2, w3, . . . , wn or W ′ = ∅ if |W | = 1.
W ′′ = w1, w2, . . . , wn−1 or W ′′ = ∅ if |W | = 1.
W ∗ = wn, wn−1, . . . , w1.
We consider the finite continued fraction associated with W to be
(2.1) [W ] = [w1, . . . , wn] = w1 +
1
w2 +
1
w3 +
1
. . . +
1
wn
TOME 74 (2024), FASCICULE 5
1812 Yining HU & Alain LASJAUNIAS
The continued fraction [W ] is a quotient of multivariate polynomials in the
variables w1, w2, . . . , wn. These polynomials are called continuants built on
W . They are defined inductively as follows:
Set ⟨∅⟩ = 1. If the sequence W has only one element, then we have ⟨W ⟩ =
W . Hence, with the above notation, the continuants can be computed,
recursively on the length |W |, by the following formula
(2.2) ⟨W ⟩ = w1⟨W ′⟩ + ⟨(W ′)′⟩ for |W | ⩾ 2.
Thus, with this notation, for any finite word W , the finite continued fraction
[W ] satisfies
(2.3) [W ] = ⟨W ⟩
⟨W ′⟩
.
It is easy to prove by induction that ⟨W ∗⟩ = ⟨W ⟩. For any finite sequences
A and B of variables over A, defining A, B as the concatenation of the
sequences, by induction on |A|, we also have the following generalization
of (2.2)
(2.4) ⟨A, B⟩ = ⟨A⟩⟨B⟩ + ⟨A′′⟩⟨B′⟩.
Using induction of |W |, we have the following classical identity
(2.5) ⟨W ⟩⟨(W ′)′′⟩ − ⟨W ′⟩⟨W ′′⟩ = (−1)|W | for |W | ⩾ 2.
3. Proof of Theorem 1.1
For n ⩾ 0, set
Wn = (σn(a))′′.
We have W0 = ∅, W1 = a, W2 = aba, etc., and |Wn| = 2n − 1.
We will prove that P ([Wn]) converges to 0 . For this, we need to following
two lemmas.
Lemma 3.1. — For all n ⩾ 0,
(3.1) Wn+1 = Wn, εn, Wn
ANNALES DE L’INSTITUT FOURIER
PERIOD-DOUBLING CONTINUED FRACTIONS ARE ALGEBRAIC 1813
where εn = a if n is even and εn = b if n is odd. In consequence, for n ⩾ 0,
we have
Wn = W ∗n .
Proof. — We prove by induction. Suppose identity (3.1) holds for n, then
Wn+2 = σ(σn+1(a))′′
= σ(Wn+1)a
= σ(Wn), σ(εn), σ(Wn)a
= Wn+1, εn+1, Wn+1.
In the last equality, we use the fact that σ(εn) = aεn+1. □
For n ⩾ 1, define un = ⟨Wn⟩ and vn = ⟨W ′n⟩, so that
[Wn] =
un
vn
.
Lemma 3.2. — For all n ⩾ 1, we have
un+1 = εnu2n,
vn+1 = εnunvn + 1.
Proof. — We use Lemma 3.1, identity (2.4), and the fact that we are in
characteristic 2.
un+1 = ⟨Wn+1⟩
= ⟨Wn, εn, Wn⟩
= ⟨Wn⟩⟨εn, Wn⟩ + ⟨W ′′n ⟩⟨Wn⟩
= ⟨Wn⟩(εn⟨Wn⟩ + ⟨W ′n⟩) + ⟨W ′′n ⟩⟨Wn⟩
= ⟨Wn⟩(εn⟨Wn⟩ + ⟨W ′n⟩) + ⟨W ′n⟩⟨Wn⟩
= ⟨Wn⟩εn⟨Wn⟩
= εnu2n.
vn+1 = ⟨W ′n+1⟩
= ⟨W ′n, εn, Wn⟩
= ⟨W ′n⟩⟨εn, Wn⟩ + ⟨(W ′n)′′⟩⟨Wn⟩
= ⟨W ′n⟩(εn⟨Wn⟩ + ⟨W ′n⟩) + ⟨(W ′n)′′⟩⟨Wn⟩
= εnunvn + ⟨W ′n⟩⟨W ′n⟩ + ⟨(W ′n)′′⟩⟨Wn⟩
= εnunvn + 1.
In the last step, we also use (2.5), and the symmetry of Wn. □
TOME 74 (2024), FASCICULE 5
1814 Yining HU & Alain LASJAUNIAS
Proof of Theorem 1.1. — For n ⩾ 1, set
Xn = Au4n + Bu3nvn + Cu2nv2n + v4n,
so that P (un/vn) = Xn/v4n.
Using Lemma 3.2 and noticing that ε2n + ab = εn(a + b) , we obtain
(3.2) Xn+1 + 1 = ε4nu4n(Xn + 1) + ε3nu4n(a + b)(1 + abu2n).
We observe that
X1 + 1 = a3(a + b) = ε0u21(a + b),
this allows us, using (3.2) and the fact that εnεn−1 = ab, to get by induction
that for all n ⩾ 1
(3.3) Xn + 1 = εn−1u2n(a + b).
Therefore
P
(
un
vn
)
= εn−1u
2
n(a + b) + 1
v4n
This is a quotient of polynomials in F2[t], and it can be easily seen from
Lemma 3.2 that the degree in t of the denominator minus that of the
numerator tends to plus infinity as n goes to infinity. And therefore the
sequence converges to 0 in F2((1/t)). □
4. Link with Riccati differential equations
In both cases, the Prouhet–Thue–Morse and the period-doubling contin-
ued fraction, a particular choice of the pair (a, b) in F2[t]2, brings us back to
an investigation undertaken 45 years ago. Indeed, in these two cases, taking
(a, b) = (t, t + 1), all the partial quotients of these continued fractions are t
or t + 1. In 1977, Baum and Sweet [2] considered all the infinite continued
fractions in F2((1/t)) whose partial quotients have all degree one.
They considered the subset D of irrational continued fractions α in
F2((1/t)) such that
α = [0; a1, a2, . . . , an, . . .]
where deg(ai) = 1 (i.e., ai = t or t + 1) for i ⩾ 1. Let P the subset of
F2((1/t)) containing all α such that |α| < 1. Hence P contains D. We have:
ANNALES DE L’INSTITUT FOURIER
PERIOD-DOUBLING CONTINUED FRACTIONS ARE ALGEBRAIC 1815
Theorem 4.1 (Baum–Sweet, 1977). — An element α ∈ P is in D if
and only if α satisfies
α2 + tα + 1 = (1 + t)β2
for some β ∈ P .
In order to obtain another formulation of Theorem 4.1, we will now use
formal differentiation in F2((1/t)). Continued fractions and formal differ-
entiation in power series fields are related, these are the main tools in
Diophantine approximation (see [5, p. 221-225]). We recall that in char-
acteristic 2, which is the case considered here, an element has derivative
zero if and only if this element is a square. Moreover the derivative of
an element is a square (hence the second derivative is zero). We have the
following theorem.
Theorem 4.2. — An element α ∈ P is in D if and only if α satisfies
(α · t(t + 1))′ = α2 + 1.
Proof. — If
α2 + tα + 1 = (1 + t)β2
then by derivation we get
(tα)′ = β2.
Consequently, we have
α2 + 1 = tα + (1 + t)(tα)′ = (α · t(t + 1))′.
Conversely, if
(α · t(t + 1))′ = α2 + 1.
then
α2 + tα + 1 = (α · t(t + 1))′ + tα = (t + 1)(α · t)′ = (t + 1)β2,
for some β ∈ P since (tα)′ is a square in P . □
Thus the Riccati differential equation
(R0) (x · t(t + 1))′ = x2 + 1
is characteristic of infinite continued fraction expansions in F2((1/t)), hav-
ing all partial quotients of degree one. Indeed, there exists a direct proof,
without Theorem 4.1, showing that a solution in F2((1/t)) of (R0) is irra-
tional and has all partial quotients of degree one (see [5, p. 225]).
TOME 74 (2024), FASCICULE 5
1816 Yining HU & Alain LASJAUNIAS
Now we turn to elements in F2((1/t)) which are algebraic over F2(t) of
degree d ⩾ 1. If α is such an element, we have (see, e.g., [5, p. 221])
(4.1) α′ = a0 + a1α + · · · + ad−1αd−1
where the coefficients ai belong to F2(t).
Concerning the algebraic continued fraction αt = CF(t) ∈ F2((1/t))
associated with the Prouhet–Thue–Morse sequence, we have observed that
it satisfies the Riccati differential equation (see [3, Proposition 2.4]):
(R) (ab(a + b)x)′ = (ab)′(1 + x2).
In fact, the continued fraction CF(p) (also algebraic of degree 4), associated
with the period-doubling sequence satisfies the same Riccati differential
equation.
Proposition 4.3. — The continued fraction αp = CF(p) satisfies the
Riccati differential equation (R)
Proof. — We know from Theorem 1.1 that
(4.2) Aα4p + Bα3p + Cα2p + 1 = 0
with
A = ab + b2 + 1, B = ab(a + b), C = ab.
Hence, by straightforword derivation of (4.2), we get
Bα′p + B′αp = A′α2p + C ′.
Therefore, observing that A′ = C ′ = (ab)′, we obtain
(ab(a + b)αp)′ = (Bαp)′ = A′α2p + C ′ = (ab)′(1 + α2p). □
Note that for the pair (a, b)=(t, t + 1), equation (R) reduces to (R0),
the one stated in Theorem 4.2. This was the way, taking in consideration
differential aspects for CF(t) and CF(p), knowing the equation (R0) in the
basic case, and inspired by the particular case published in [4], that allowed
us to guess the coefficients for the algebraic equation satisfied by CF(p).
BIBLIOGRAPHY
[1] J.-P. Allouche & J. Shallit, Automatic sequences. Theory, Applications, Gener-
alizations, Cambridge University Press, 2003, xvi+571 pages.
[2] L. E. Baum & M. M. Sweet, “Badly approximable power series in characteristic 2”,
Ann. Math. 105 (1977), no. 3, p. 573-580.
[3] Y. Bugeaud & G.-N. Han, “The Thue–Morse continued fractions in characteristic
2 are algebraic”, to be published in Acta Arithmetica, 2023.
[4] Y. Hu & G.-N. Han, “On the algebraicity of Thue–Morse and period-doubling con-
tinued fractions”, Acta Arith. 203 (2022), no. 4, p. 353-381.
ANNALES DE L’INSTITUT FOURIER
PERIOD-DOUBLING CONTINUED FRACTIONS ARE ALGEBRAIC 1817
[5] A. Lasjaunias, “A survey of Diophantine approximation in fields of power series”,
Monatsh. Math. 130 (2000), no. 3, p. 211-229.
[6] ——— , “Continued fractions”, https://arxiv.org/abs/1711.11276, 2017.
Manuscrit reçu le 6 avril 2022,
révisé le 20 septembre 2022,
accepté le 17 novembre 2022.
Yining HU
School of Mathematics and Statistics,
Huazhong University of Science and Technology,
Wuhan (P. R. China)
Current address:
Institute for Advanced Study in Mathematics,
Harbin Institute of Technology,
Harbin (P. R. China)
huyining@protonmail.com
Alain LASJAUNIAS
Mathematical Institute Bordeaux University,
Bordeaux (France)
lasjauniasalain@gmail.com
TOME 74 (2024), FASCICULE 5


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