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On the automaticity of sequences defined by continued fractions

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Pith's one-line read The paper proves that the Thue-Morse and period-doubling continued fractions are congruent modulo 4 to explicit algebraic power series, so their coefficient sequences modulo 4 are 2-automatic.

desk verdict A real new result on automaticity of continued fractions; the reader's rejection is based on a misreading of S_m, and the main induction survives. read the letter →

arxiv 1908.02384 v2 pith:P5OW6KNO submitted 2019-08-06 math.CO math.NT

Guoniu Han , Yining Hu This is my paper · ORCID
classification math.COmath.NT MSC 11B8511J7011B5011Y6505A15
keywords automaticsequencecontinuedfractionThue-Morseperiod-doublingHankeldeterminant2-automaticformalpowerseriescongruencemodulo4
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper studies the converse of the usual automatic-sequence/continued-fraction question: instead of asking what continued fractions automatic sequences produce, it takes two continued fractions whose partial quotients are the automatic Thue-Morse and period-doubling sequences and asks what power series they define. The main claim is that the two series C(x) and D(x) are congruent modulo 4 to explicit algebraic series over Z, so by the Denef-Lipschitz theorem their coefficient sequences modulo 4 are 2-automatic, meaning each coefficient can be generated by a finite automaton reading the index in binary. The proof is a direct calculation: the authors guess closed forms for certain subsequences of the canonical convergents P_n/Q_n, prove them by an induction involving eight coupled subsequences, then pass to the limit to identify C(x) and D(x) modulo 4. If correct, these are concrete examples of formal power series that are algebraic modulo a prime power while being generated by automatic continued fractions.

What carries the argument

The load-bearing object is the pair (P_n(x),Q_n(x)) of canonical convergents of the Stieltjes continued fraction, with C(x)=lim P_n/Q_n, together with eight coupled subsequences P_{2^k-1}, P_{2^k-2}, Q_{2^k-1}, Q_{2^k-2} expressed modulo 4 in terms of truncated lacunary sums S_m=sum_{j=0}^{m-1} $x^{{2^j}}$, S^e_m=sum $x^{{2^{2j}}$}, and S^o_m=sum $x^{{2^{2j+1}}$} (Proposition 3.2). The induction over these eight subsequences produces the explicit limit series; the Cartier operators Lambda_0 and Lambda_1 then convert that series into a finite 2-kernel, proving automaticity directly. The contraction theorem for Stieltjes and Jacobi continued fractions turns the Thue-Morse congruence into the period-doubling congruence, and the same explicit series feeds the degree-4 annihilator and Hankel determinant arguments.

What would settle it

Compute the first convergents directly from the recurrence: P_0=P_1=Q_0=1, Q_1=1-x, P_2=1-x, Q_2=1-2x, Q_3=1-x-$x^{2}$. Proposition 3.2 items 3), 5), and 7) for m=1 give expressions such as -1+$x^{{-1}}$, 1+x+$2x^{2}$, and 3, which do not match these polynomials. A reader can check this disagreement in two lines and thereby determine whether the announced congruences are supported.

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Extended reading notes

Core claim

The central discovery is Theorem 1.1 and Theorem 1.2: with C(x)=sum c_n x^n the Thue-Morse Stieltjes continued fraction and D(x)=sum d_n x^n the period-doubling one, the paper proves the congruences C(x) equivalent to ($\sqrt$(1-4x)-1)/(2x)+1+$\sqrt$(2 $\sqrt$(1-4x)-1) modulo 4 and D(x) equivalent to ((1+$\sqrt$(1+4x))($\sqrt$(2 $\sqrt$(1-$4x^{2}$)-1)-2))/(2x) modulo 4. As a direct corollary, via the Denef-Lipschitz theorem, the reduced coefficient sequences (c_n mod 4) and (d_n mod 4) are 2-automatic, and the paper explicitly computes their 2-kernels, which have 9 states and 5 states respectively. The paper further shows that both reduced series satisfy the degree-4 polynomial S(x,y)=($xy^{2}$+y+1)^2 in Z/4Z[x,y], that no lower-degree polynomial with invertible leading coefficient in the Laurent series ring annihilates either series, and that the Hankel determinant sequences of C(x) and D(x) are 2-automatic.

Load-bearing premise

The explicit closed-form formulas in Proposition 3.2 for the convergent subsequences P_{2^k-1}, P_{2^k-2}, Q_{2^k-1}, Q_{2^k-2}, claimed for all m >= 1, are the load-bearing step; the base case m=1 already conflicts with the recurrence defining P_n and Q_n, and if these formulas are not true, the limit computation of C(x), the congruences (1.3) and (1.4), and the subsequent automaticity results do not follow.

Editorial extensions

If this is right

  • The coefficient sequences modulo 4 are 2-automatic, with explicit transducers of 9 states for (c_n mod 4) and 5 states for (d_n mod 4).
  • Both reduced series are algebraic of degree 4 over Z/4Z[x], sharing the same annihilating polynomial (xy^2+y+1)^2 and admitting no lower-degree annihilator with invertible leading Laurent-series coefficient.
  • The Hankel determinant sequences of C(x) and D(x) are 2-automatic.
  • The contraction identity C(x)=1/(1-x-x^2 D(-x^2)) makes the period-doubling theorem a formal consequence of the Thue-Morse theorem, so the two automaticity results stand or fall together.
  • These examples show that automatic continued fractions can define series that are algebraic modulo a prime power, in contrast with Bugeaud's theorem for real algebraic numbers of degree at least three.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • One testable extension is to run the same eight-subsequence scheme on the m=3 kernels reported in the paper; if the induction base is repaired or re-indexed, the m=3 automaticity could likely be proved directly, giving the first case beyond modulo 4 of Conjecture 1.9.
  • The explicit 2-kernels are self-contained objects: they let a reader generate coefficients modulo 4 by iterating a two-letter transducer without computing continued fractions, so any finite prefix of (c_n mod 4) or (d_n mod 4) can be checked independently against the algebraic congruences.
  • If the closed forms in Proposition 3.2 are adjusted to hold for m >= 1, the same limit argument would go through unchanged, since the infinite-series identity is what the induction ultimately approximates.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

1 major / 5 minor

Summary. The paper studies two formal Stieltjes continued fractions C(x) and D(x) whose coefficients are respectively the Thue-Morse and period-doubling sequences. It claims that modulo 4 each of these series is congruent to an explicit algebraic power series (Theorems 1.1 and 1.2) and consequently, by the Denef-Lipschitz theorem, that the sequences of coefficients of their power series expansions are 2-automatic. The main technical work is Proposition 3.2, which gives explicit formulas for eight subsequences of the numerator and denominator polynomials of the convergents of C(x); these formulas are proved by an induction and then passed to the limit. The paper also computes explicit 2-kernels for the reduced coefficient sequences, proves a lower bound on the algebraic degree modulo 2 (Theorem 1.6), and derives the 2-automaticity of the corresponding Hankel determinants.

Significance. If the proofs are correct, this is a substantial contribution to the converse direction of the continued-fraction/automatic-sequence literature: automatic sequences are used as partial quotients, and the resulting generating series are shown to be algebraic modulo a fixed prime power. The paper gives explicit, parameter-free algebraic closed forms and carries out the Cartier-operator computations that exhibit finite kernels. The proof of Proposition 3.2 is intricate and is the load-bearing part of the argument; the period-doubling case is then handled by a contraction argument rather than by repeating the induction. The algebraic-degree result and the Hankel-determinant corollaries are natural and nontrivial additions.

major comments (1)
  1. [§3, Proposition 3.2] The alleged base-case failure of Proposition 3.2 does not occur. With the intended reading of the exponent in item 5 as 2^{2m-1} rather than 2^{2m}-1, the m=1 values are P_2=1-x, Q_2≡1-2x, and Q_3≡1-x-x^2 modulo 4, all matching the recurrence of Lemma 3.1. The same reading is forced by the induction step in the proof of item 5. I therefore see no load-bearing gap in the derivation of congruence (1.3).
minor comments (5)
  1. [§3, Proposition 3.2] The exponent notation in Proposition 3.2 is very easy to misread. In particular, item 5 should be typeset with explicit braces, e.g. 2x^{2^{2m-1}}, because the unbraced rendering can be read as 2x^{2^{2m}-1}. I recommend adding parentheses or braces to all exponents in this proposition.
  2. [§3, Proposition 3.3] The sentence beginning 'The constant term of Q_{2^{2m}+2}(x) being 1' appears to be a typo: the next line uses Q_{2^{2m}-2}(x), and the constant term of that denominator is indeed 1.
  3. [§4, proof of Theorem 1.2] In the sentence 'Then our goal (1.4) can be written as D(x) ≡ (H_1(x)H_3(x)-1)/x', the factor H_1 should be H_2; the subsequent calculation correctly uses H_2(-x^2)H_3(-x^2).
  4. [§5] In the irreducibility check for y^4+y+x over F_2[x], the statement that a linear factor must be y+x or y+1 is too quick. The standard degree argument is that a root a ∈ F_2[x] would satisfy a^4+a=x, which is impossible by degree comparison; please include that justification.
  5. [§1] Minor language: 'The right hand side of congruence (1.3) and (1.4) are respectively' should be 'is respectively'; also, the displayed polynomials in Section 5 would benefit from consistent notation such as x^1 and x^2 instead of x1 and x2.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the paper proves its explicit formulas by induction and derives automaticity from external Denef-Lipschitz and direct kernel calculations.

full rationale

I walked the derivation chain. The central claims are Theorems 1.1 and 1.2, which give closed-form congruences for the Thue-Morse and period-doubling continued fractions modulo 4. These are not assumed or fitted: the explicit subsequence formulas in Proposition 3.2 are proved by induction from the recurrence in Lemma 3.1, which is derived directly from the definition of the Thue-Morse sequence and the continued-fraction identities (2.3)-(2.4). Proposition 3.3 then takes the limit of the proved convergent subsequences to obtain the power-series expression for C(x), and Theorem 1.1 is an algebraic rewriting of that expression using Lemma 3.4. Theorem 1.2 derives D(x) from C(x) via the Contraction Theorem, an external classical result, and the subsequent automaticity statements follow either from the external Denef-Lipschitz theorem or from explicit finite 2-kernel computations in the proofs of Theorems 1.4 and 1.5. The only self-citation that appears in a load-bearing position is the reference to [22] for Lemma 3.4, the congruence sqrt(1-4x) = 1 + 2*sum_{k>=1} x^{2^k} mod 4. This is a parameter-free elementary identity that does not involve the target continued fractions, is independently verifiable, and is not equivalent to the paper's conclusions, so under the review rules it counts as real evidence rather than circularity. No curve fitting, no fitted parameter renamed as a prediction, no imported uniqueness theorem, and no ansatz smuggled in by citation occurs. The possible base-case objection raised by the Reader concerns the truth of the Proposition 3.2 formulas, which is a correctness question, not a circularity question; even if the induction had a gap, the argument is not circular because it does not assume the target congruence. Accordingly, the circularity score is 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No fitted parameters or invented entities. The central dependencies are standard theorems plus the paper's own Proposition 3.2. Because Proposition 3.2 is false as stated, the proof of the main claim is unsupported.

assumptions (4)
  • standard math Denef-Lipschitz theorem: algebraic power series over Z_p have p-automatic coefficient sequences modulo p^alpha.
    Quoted as Theorem 1.3 and used to conclude automaticity of (cbar_n) and (dbar_n) from Theorems 1.1 and 1.2.
  • standard math Convergence of Stieltjes continued fractions in Z[[x]] (Theorem 2.1).
    Used to take limits P_n/Q_n and identify C(x) and D(x).
  • standard math Contraction theorem linking Stieltjes and Jacobi continued fractions (Theorem 2.3).
    Used in the proof of Theorem 1.2 to relate C(x) and D(x).
  • ad hoc to paper The explicit subsequence formulas of Proposition 3.2 for P_{2^k-1}, P_{2^k-2}, Q_{2^k-1}, and Q_{2^k-2}.
    These guessed formulas are the load-bearing computational lemma. The induction is invalid because the stated m=1 base cases contradict the recurrence: P2=1-x, Q2=1-2x, Q3=1-x-x^2, while the formulas contain x^{-1} terms that cannot appear in polynomials.

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Pith. "Pith review of On the automaticity of sequences defined by continued fractions." pith.science (2026). https://pith.science/paper/P5OW6KNO

@misc{pith2026190802384,
  author       = {Pith},
  title        = {Pith review of: On the automaticity of sequences defined by continued fractions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/P5OW6KNO}},
  note         = {Machine review of arXiv:1908.02384}
}
abstract

Continued fraction expansions and Hankel determinants of automatic sequences are extensively studied during the last two decades. These studies found applications in number theory in evaluating irrationality exponents. The present paper is motivated by the converse problem: to study continued fractions of which the elements form an automatic sequence. We consider two such continued fractions defined by the Thue-Morse and period-doubling sequences respectively, and prove that they are congruent to algebraic series in $\mathbb{Z}[[x]]$ modulo $4$. Consequently, the sequences of the coefficients of the power series expansions of the two continued fractions modulo $4$ are $2$-automatic. Our approach is to first guess the explicit formulas of certain subsequences of $(P_n(x))$ and $(Q_n(x))$, where $P_n(x)/Q_n(x)$ is the canonical representation of the truncated continued fractions, then prove these formulas by an intricate induction involving eight subsequences while exploiting the relations between these subsequences.

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