arXiv:2608.15119v1

Transcendence of continued fractions over function fields and a quantitative version of Uchiyama's theorem

Federico Accossato, Nadir Murru, Giuliano Romeo, Giulia Salvatori

math.NT11J7011D8811J81

Paper metadata

Original submission: August 15, 2026
Claimed MSC: 11J70, 11D88, 11J81
arXiv categories: math.NT
Paper license: CC BY-SA 4.0 ↗
Comments: 20 pages, comments are welcome!

Abstract

Given a field $K$ , let $K((T^{-1}))$ be the field of formal power series. Continued fractions in $K((T^{-1}))$ can be defined by analogy with classical real continued fractions and have been widely studied. Some results establish the transcendence of elements of $K((T^{-1}))$ arising from special families of continued fractions, but much remains to be explored. In this paper, assuming that $K$ has characteristic zero, we improve the known analogues of the Maillet--Baker criteria for quasi-periodic continued fractions. A central tool that we prove is a quantitative version of Uchiyama's analogue of Roth's theorem in function fields, which gives an explicit bound for the number of exceptionally good rational approximations to an algebraic power series. This quantitative estimate also yields a Davenport--Roth-type upper bound on the growth of the denominators of the convergents of algebraic elements. Finally, we prove that palindromic continued fractions are either quadratic or transcendental, as in the real case, but using a different proof strategy.

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Audited against arXiv v1

Not a correctness certificate. A “Correct” result may include yellow typos or minor formal corrections that do not affect substantive soundness. It means this audit found no unresolved substantive error under the stated criteria; it does not replace expert scrutiny or formal verification.

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Detailed mathematical audit

Generated August 18, 2026

01Statements4 reported findingsCorrect

The quantitative function-field Roth bound, the Davenport–Roth denominator-growth estimate, both quasi-periodic transcendence criteria, and the palindrome criterion are correct over characteristic-zero coefficient fields.

Theorem 6Correct

Exceptionally good rational approximations have the claimed quantitative bound

Pages 3–4 and 7–10 · Theorems 6 and 16, Proposition 17 · arXiv:2608.15119v1

Solutions are counted modulo simultaneous multiplication by $K^\times$ , as specified in Remark 7. For an integral algebraic Laurent series of degree $d$ , the explicit choice $m=\lfloor100d^2\varepsilon^{-2}\rfloor+1$ and the displayed $\delta$ satisfy Uchiyama's separation hypotheses. Ordering reduced solutions by denominator norm and spacing their indices produces $m$ forbidden approximants if more than $k+(m-1)\ell$ occur; the estimates for $k$ and $\ell$ give $\exp(Cd^2\varepsilon^{-2})$ . Multiplying a general algebraic series by its leading coefficient makes it integral, and the finite split by $\gcd(a,q)$ preserves the same exponential order.

Theorem 18Correct

Algebraic denominator growth satisfies the Davenport–Roth bound

Pages 10–12 · Theorem 18 · arXiv:2608.15119v1

Writing $\gamma_j=\deg q_{j+1}/\deg q_j$ , Liouville's inequality gives a uniform upper bound for $\gamma_j$ . Indices with $\gamma_j>1+\varepsilon$ yield distinct solutions of Theorem 6, so there are fewer than $\exp(c\varepsilon^{-2})$ of them; all other indices contribute at most $i\varepsilon$ to $\log\deg q_i$ . Choosing $\varepsilon$ proportional to $1/\sqrt{\log i}$ gives $\log\log|q_i|<\frac{Ci}{\sqrt{\log i}},$ with finitely many small indices absorbed into $C$ .

Theorem 4Correct

Both quasi-periodic criteria force quadraticity or transcendence

Pages 3 and 12–17 · Theorem 4 and Section 4.1 · arXiv:2608.15119v1

In part (a), the periodic quadratic approximants share $n_i+\lambda_i r_i$ initial partial quotients with $\alpha$ . The resulting upper approximation bound, the quadratic Liouville lower bound, and Theorem 18 contradict the limsup hypothesis if an algebraic $\alpha$ has degree greater than two. In part (b), Lemma 19 bounds the height of the quadratic approximants. The two applications of the function-field Subspace Theorem either force a repeated quadratic approximant or a rational linear dependence whose limiting ratio gives an approximation violating Liouville's inequality. Both alternatives exclude algebraic degree at least three.

Theorem 5Correct

Arbitrarily long palindromic prefixes imply quadraticity or transcendence

Pages 3 and 17–18 · Theorem 5 and its proof · arXiv:2608.15119v1

For every palindromic prefix, symmetry of the convergent matrix gives $p_n=q_{n-1}$ . The three displayed linear forms at $(p_{n-1},q_n,q_{n-1})$ have product smaller than a negative power of the vector height, so Ratliff's Subspace Theorem puts infinitely many such vectors in one proper rational subspace. Passing to the limit in that fixed relation yields $u\alpha^2+v\alpha+w=0.$ Thus an algebraic $\alpha$ is quadratic; otherwise it is transcendental.

02Proofs4 reported findingsCorrect

The quantitative parameter check, reduction from general to integral algebraic series, denominator-growth decomposition, quadratic-approximant arguments, and two Subspace-Theorem applications are correct and complete. The material Uchiyama and Ratliff inputs match the primary sources.

Theorem 16 and Theorem 6Correct and complete

The quantitative Uchiyama argument and the integrality reduction close all cases

Pages 7–10 · Theorem 16, Proposition 17, and proof of Theorem 6 · arXiv:2608.15119v1

The chosen parameters meet each of Uchiyama's conditions (3)–(5), including positivity of the relevant denominator in the nontrivial case. Consecutive reduced fractions of equal denominator norm cannot both satisfy the approximation inequality, so the ordered list has strictly increasing norms. The $k$ and $\ell$ selections then force the exact forbidden configuration from Uchiyama's Section 7 and have the required exponential bound. For nonintegral $\alpha$ , each monic divisor $b$ of the leading coefficient gives a coprime integral approximation after removing $b$ ; Theorem 16 handles large denominators and Proposition 17 bounds the finitely many smaller norms. Summing over the finitely many $b$ completes the proof.

Uchiyama, Rational approximations to algebraic functions ↗

Theorem 18Correct and complete

The large-jump and small-jump denominator estimates are compatible

Pages 10–12 · proof of Theorem 18 · arXiv:2608.15119v1

Every large ratio $\gamma_j>1+\varepsilon$ turns the $j$ th convergent into a solution with exponent $2+\varepsilon$ , and distinct convergents remain distinct modulo $K^\times$ . Theorem 6 therefore bounds their count. Liouville's estimate bounds the size of each large contribution, while the remaining logarithms are at most $\varepsilon$ . The final choice of $\varepsilon$ keeps the exponentially bounded exceptional term below the desired order, and all constants depend only on $\alpha$ .

Theorem 4(a)Correct and complete

Periodic quadratic approximants and denominator growth give the required contradiction

Pages 12–14 · proof of Theorem 4(a) · arXiv:2608.15119v1

The purely periodic block produces a reduced quadratic series, and the shared-prefix formula gives the stated approximation upper bound. Lemmas 13–14 supply the matching height and conjugate estimates with their parameter dependence intact. Under algebraic degree greater than two, the quadratic Liouville inequality then bounds $\lambda_i$ through the degrees of convergent denominators. Applying Theorem 18 and $r_i<Cn_i$ yields a finite bound for the limsup in (1), which is the required contradiction.

Theorems 4(b) and 5Correct and complete

The Subspace-Theorem applications verify the remaining transcendence criteria

Pages 14–18 · proofs of Theorem 4(b) and Theorem 5 · arXiv:2608.15119v1

In Theorem 4(b), the limsup condition supplies a subsequence on which the approximation dominates every required height power. The separate $v_i=0$ case, the three-form case, and the possible vanishing coefficient in the limiting relation are all treated before Liouville is invoked. In Theorem 5, the three forms are linearly independent, their product estimate uses exactly the palindrome identity, and unbounded vector height activates the Subspace Theorem. The resulting fixed subspace relation survives the limit and gives the claimed quadratic equation.

Ratliff, The Thue–Siegel–Roth–Schmidt theorem for algebraic functions ↗

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Paper: arXiv:2608.15119v1
Authors listed: Federico Accossato, Nadir Murru, Giuliano Romeo, Giulia Salvatori
Audit date: August 18, 2026

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Transcendence of continued fractions over function fields and a quantitative version of Uchiyama's theorem

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

Exceptionally good rational approximations have the claimed quantitative bound

Algebraic denominator growth satisfies the Davenport–Roth bound

Both quasi-periodic criteria force quadraticity or transcendence

Arbitrarily long palindromic prefixes imply quadraticity or transcendence

The quantitative Uchiyama argument and the integrality reduction close all cases

The large-jump and small-jump denominator estimates are compatible

Periodic quadratic approximants and denominator growth give the required contradiction

The Subspace-Theorem applications verify the remaining transcendence criteria

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