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This note is a complement to an article which was published, six years ago, in The Ramanujan Journal (vol. 45.3, 2018). Here, the goal is to fully describe a singular transcendental continued fraction in Q((T^-1)), tied to a particular infinite two letters word.

This short note is a comment on a historical aspect of a famous formula dating from the 18th century.

By replacing the letters to polynomials in F_2[t], an infinite word, over a finite alphabet, can be seen as the sequence of partial quotients of a continued fraction in F_2((1/t)). Here is described a family of such infinite words, corresponding to continued fractions which are algebraic over F_2(t). This family includes a classical example already studied in different previous works by Y. Hu, G-N. Han, Y. Bugeaud and the author of this note.

Considering an arbitrary pair of distinct and non constant polynomials, $a$ and $b$ in $\mathbb{F}_2[t]$, we build a continued fraction in $\mathbb{F}_2((1/t))$ whose partial quotients are only equal to $a$ or $b$. In a previous work of the first author and Han (to appear in Acta Arithmetica), 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 fractions 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 (arXiv:2203.02213) 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.

In different areas of discrete mathematics, a certain type of polynomials, having coefficients in a field K of finite characteristic, has been considered. The form and the degree of these polynomials, here called projective, are simply linked to the characteristic p of K. Roots of these projective polynomials are particular algebraic elements over K, called hyperquadratic. For a general algebraic element of degree d over K, we discuss the possibility of being hyperquadratic. Using a method of differential algebra, we obtain, for particular fields K = Fp, projective polynomials only having polynomial factors of degree 1 or 2.

We discuss the form of certain algebraic continued fractions in the field of power series over $F_p$, where p is an odd prime number. This leads to give explicit continued fractions in these fields, satisfying an explicit algebraic equation of arbitrary degree $d\geq 2$ and having an irrationality measure equal to $d$. Our results are based on a mysterious finite sequence of rational numbers.

We present a general introduction to continued fractions, with special consideration to the function fields case. These notes were prepared for a summer class given this year in Beijing at Beihang university.

Given an odd prime number p, we describe a continued fraction in the field F(p) of power series in 1/T with coefficients in the finite field F_p, where T is a formal indeterminate. This continued fraction satisfies an algebraic equation of a particular type, with coefficients in F_p[T] which are explicitely given. We observe the close connection with other algebraic continued fractions studied thirty years ago by Mills and Robbins.

The goal of this survey paper is to present, in chronological order, certain research works on continued fractions in power series fields over a finite field, all of them being derivated from some examples introduced thirty years ago by Mills and Robbins., Comment: 6 pages

We explicitly describe a noteworthy transcendental continued fraction in the field of power series over Q, having irrationality measure equal to 3. This continued fraction is a generating function of a particular sequence in the set {1, 2}. The origin of this sequence, whose study was initiated in a recent paper, is to be found in another continued fraction, in the field of power series over $\mathbb{F}\_3$, which satisfies a simple algebraic equation of degree 4, introduced thirty years ago by D. Robbins.

In this work we extend our study on a link between automaticity and certain algebraic power series over finite fields. Our starting point is a family of sequences in a finite field of characteristic $2$, recently introduced by the first author in connection with algebraic continued fractions. By including it in a large family of recurrent sequences in an arbitrary finite field, we prove its automaticity. Then we give a criterion on automatic sequences, generalizing a previous result and this allows us to present new families of automatic sequences in an arbitrary finite field., Comment: 10 pages

In this note, we describe a family of particular algebraic, and nonquadratic, power series over an arbitrary finite field of characteristic 2, having a continued fraction expansion with all partial quotients of degree one. The main purpose being to point out a common origin with another analogue family in odd characteristic, which has been studied in [LY1]., Comment: 6 pages

The aim of this note is to show the existence of a correspondance between certain algebraic continued fractions in fields of power series over a finite field and automatic sequences in the same finite field. this connection is illustrated by three families of examples and a counterexample.

Casually introduced thirty years ago, a simple algebraic equation of degree 4, with coefficients in Fp[T], has a solution in the field of power series in 1/T, over the finite field Fp. For each prime p > 3, the continued fraction expansion of this solution is remarkable and it has a different general pattern according to the remainder, 1 or 2, in the division of p by 3. We describe two very large families of algebraic continued fractions, each containing these solutions, according to the class of p modulo 3. We can compute the irrationality measure for these algebraic continued fractions and, as a consequence, we obtain two different values for the solution of the quartic equation, only depending on the class of p modulo 3., Comment: 21 pages, 2 figures

In 1986, some examples of algebraic, and nonquadratic, power series over a finite prime field, having a continued fraction expansion with partial quotients all of degree one, were discovered by W. Mills and D. Robbins. In this note we show how these few examples are included in a very large family of continued fractions for certain algebraic power series over an arbitrary finite field of odd characteristic.

The first part of this note is a short introduction on continued fraction expansions for certain algebraic power series. In the last part, as an illustration, we present a family of algebraic continued fractions of degree 4, including a toy example considered about thirty years ago in a pioneer work in this area.

We are concerned with power series in 1/T over a finite field of 3 elements $\F_3$. In a previous article, Alain Lasjaunias investigated the existence of particular power series of elements algebraic over $\F_3[T]$, having all partial quotients of degree 1 in their continued fraction expansion. Here, we generalize his result and we make a conjecture about the elements with all partial quotients of degree 1, except maybe the first ones.

In 1985, Robbins observed by computer the continued fraction expansion of certain algebraic power series over a finite field. Incidentally, he came across a particular equation of degree 4 in characteristic p=13. This equation has an analogue for all primes p>=5. There are two patterns for the continued fraction of the solution of this equation, according to the residue of p modulo 3. We describe this pattern in the first case, considering especially p=7 and p=13. in the second case we only give indications.

We report thermal transport measurements performed on the quasi-one-dimensional Bechgaard salts (TMTSF)2ClO4 and (TMTSF)2PF6 along the a-direction. For both salts, magnon-drag effects are found to contribute considerably to thermopower above 80 K. These results imply spin-charge separation in the metallic state for both salts. Moreover, a linear temperature-dependent thermal conductivity is found to be unaffected by anion disorder below an anion ordering transition temperature TAO~25 K in (TMTSF)2ClO4., Comment: 12 pages, 3 figures and 1 table

There exists a particular subset of algebraic power series over a finite field which, for different reasons, can be compared to the subset of quadratic real numbers. The continued fraction expansion for these elements, called hyperquadratic, can sometimes be made explicit. Here we describe this expansion for a wide family of hyperquadratic power series in odd characteristic. This leads to consider interesting recurrent sequences in the finite base field when it is not a prime field.

We have investigated the effect of the magnetic field (B) on the very low-temperature equilibrium heat capacity ceq of the quasi-1 D organic compound (TMTTF)2Br, characterized by a commensurate Spin Density Wave (SDW) ground state. Below 1K, ceq is dominated by a Schottky-like contribution, very sensitive to the experimental time scale, a property that we have previously measured in numerous DW compounds. Under applied field (in the range 0.2- 7 T), the equilibrium dynamics, and hence ceq extracted from the time constant, increases enormously. For B = 2-3 T, ceq varies like B2, in agreement with a magnetic Zeeman coupling. Another specific property, common to other Charge/Spin density wave (DW) compounds, is the occurrence of metastable branches in ceq, induced at very low temperature by the field exceeding a critical value. These effects are discussed within a generalization to SDWs in a magnetic field of the available Larkin-Ovchinnikov local model of strong pinning. A limitation of the model when compared to experiments is pointed out., Comment: 10 pages, 11 figures

Equilibrium heat relaxation experiments provide evidence that the ground state of the commensurate spin density wave (SDW) compound (TMTTF)$_2$Br after the application of a sufficient magnetic field is different from the conventional ground state. The experiments are interpreted on the basis of the local model of strong pinning as the deconfinement of soliton-antisoliton pairs triggered by the Zeeman coupling to spin degrees of freedom, resulting in a magnetic field induced density wave glass for the spin carrying phase configuration., Comment: 4 pages, 5 figures

We briefly review different approaches used recently to describe collective effects in the strong pinning model of disordered charge and spin density waves, in connection with the CRTBT very low temperature heat relaxation experiments., Comment: 4 pages, invited talk at ECRYS-2005

We show that the low temperature ($T<0.5$ K) time dependent non-exponential energy relaxation of quasi-one-dimensional (quasi-1D) compounds strongly differ according to the nature of their modulated ground state. For incommensurate ground states, such as in (TMTSF)$_2$PF$_6$ the relaxation time distribution is homogeneously shifted to larger time when the duration of the heat input is increased, and exhibits in addition a scaling between the width and the position of the peak in the relaxation time distribution, $w^{2}\sim\ln{(\tau_{m})}$. For a commensurate ground state, as in (TMTTF)$_2$PF$_6$, the relaxation time spectra show a bimodal character with a weight transfer between well separated slow and fast entities. Our interpretation is based on the dynamics of defects in the modulated structure, which depend crucially on the degree of commensurability., Comment: 4 pages, 4 figures

We investigate collective effects in the strong pinning model of disordered charge and spin density waves (CDWs and SDWs) in connection with heat relaxation experiments. We discuss the classical and quantum limits that contribute to two distinct contribution to the specific heat (a $C_v \sim T^{-2}$ contribution and a $C_v \sim T^{\alpha}$ contribution respectively), with two different types of disorder (strong pinning versus substitutional impurities). From the calculation of the two level system energy splitting distribution in the classical limit we find no slow relaxation in the commensurate case and a broad spectrum of relaxation times in the incommensurate case. In the commensurate case quantum effects restore a non vanishing energy relaxation, and generate stronger disorder effects in incommensurate systems. For substitutional disorder we obtain Friedel oscillations of bound states close to the Fermi energy. With negligible interchain couplings this explains the power-law specific heat $C_v \sim T^{\alpha}$ observed in experiments on CDWs and SDWs combined to the power-law susceptibility $\chi(T)\sim T^{-1+\alpha}$ observed in the CDW o-TaS$_3$., Comment: 13 pages, 10 figures, improvements in the presentation

We present the first low-temperature thermodynamic investigation of the controlled physisorption of He$^{4}$ gas in carbon single-wall nanotube (SWNT) samples. The vibrational specific heat measured between 100 mK and 6 K demonstrates an extreme sensitivity to outgassing conditions. For bundles with a few number of NTs the extra contribution to the specific heat, C$_{ads}$, originating from adsorbed He$^{4}$ at very low density displays 1D behavior, typical for He atoms localized within linear channels as grooves and interstitials, for the first time evidenced. For larger bundles, C$_{ads}$ recovers the 2D behaviour akin to the case of He$^{4}$ films on planar substrates (grafoil)., Comment: 4 pages, 3 figures, submitted to Phys. Rev. Lett

Measurements of the magnetic susceptibility and its anisotropy in the quasi-one-dimensional system o-TaS_{3} in its low-T charge density wave (CDW) ground state are reported. Both sets of data reveal below 40 K an extra paramagnetic contribution obeying a power-law temperature dependence \chi(T)=AT^{-0.7}. The fact that the extra term measured previously in specific heat in zero field, ascribed to low-energy CDW excitations, also follows a power law C_{LEE}(0,T)=CT^{0.3}, strongly revives the case of random exchange spin chains. Introduced impurities (0.5% Nb) only increase the amplitude C, but do not change essentially the exponent. Within the two-level system (TLS) model, we estimate from the amplitudes A and C that there is one TLS with a spin s=1/2 localized on the chain at the lattice site per cca 900 Ta atoms. We discuss the possibility that it is the charge frozen within a soliton-network below the glass transition T_{g}~40 K determined recently in this system., Comment: 7 pages, 3 figures, submitted to Europhysics Letters

We report on extended investigation of the thermal transport and acoustical properties on hard carbon samples obtained by pressurization of C60 fullerene. Structural investigations performed by different techniques on the same samples indicate a very inhomogeneous structure at different scales, based on fractal-like amorphous clusters on the micrometer to submillimetre scale, which act as strong acoustic scatterers, and scarce microcrystallites on the nanometer scale. Ultrasonic experiments show a rapid increase in the attenuation with frequency, corresponding to a decrease in the localization length for vibrations. The data give evidence for a crossover from extended phonon excitations to localized fracton excitations. The thermal conductivity is characterized by a monotonous increase versus temperature, power law T^(1.4), for T ranging from 0.1 to 10 K, without any well-defined plateau, and a strictly linear-in-T variation between 20 and 300K. The latter has to be related to the linear-in-T decrease of the sound velocity between 4 and 100 K, both linear regimes being characteristic of disordered or generally aperiodic structures, which can be analysed by the "phonon-fracton hopping" model developed for fractal and amorphous structures., Comment: 30 pages, 8 EPS figures, 4 JPG pictures

We show that the dynamics of disordered charge density waves (CDWs) and spin density waves (SDWs) is a collective phenomenon. The very low temperature specific heat relaxation experiments are characterized by: (i) ``interrupted'' ageing (meaning that there is a maximal relaxation time); and (ii) a broad power-law spectrum of relaxation times which is the signature of a collective phenomenon. We propose a random energy model that can reproduce these two observations and from which it is possible to obtain an estimate of the glass cross-over temperature (typically $T_g \simeq 100 - 200$ mK). The broad relaxation time spectrum can also be obtained from the solutions of two microscopic models involving randomly distributed solitons. The collective behavior is similar to domain growth dynamics in the presence of disorder and can be described by the dynamical renormalization group that was proposed recently for the one dimensional random field Ising model [D.S. Fisher, P. Le Doussal and C. Monthus, Phys. Rev. Lett. {\bf 80}, 3539 (1998)]. The typical relaxation time scales like $\tau^{\rm typ} \sim \tau_0 \exp{(T_g/T)}$. The glass cross-over temperature $T_g$ related to correlations among solitons is equal to the average energy barrier and scales like $T_g \sim 2 x \xi_0 \Delta$. $x$ is the concentration of defects, $\xi_0$ the correlation length of the CDW or SDW and $\Delta$ the charge or spin gap., Comment: 20 pages, 16 figures