arXiv:2607.24427v1

On $C^k$ -functions mapping $\mathbb{Q}$ into itself and Mahler's problem on Liouville numbers

Jean Lelis, Carlos Gustavo Moreira, Elaine Silva

math.NT11J8111J91

Paper metadata

Original submission: July 27, 2026
Claimed MSC: 11J81, 11J91
arXiv categories: math.NT
Paper license: arXiv non-exclusive distribution license ↗
Comments: 11 pages

Abstract

Liouville numbers form a classical class of transcendental real numbers characterized by exceptionally strong rational approximations. A theorem of Maillet shows that non-constant rational functions with rational coefficients preserve the Liouville property, motivating a question of Mahler on whether analogous phenomena hold for transcendental functions. In this paper, we address this problem for real functions of finite smoothness. For any $\varepsilon>0$ , we construct an uncountable set of $C^k$ -functions on $\mathbb{R}$ , dense with respect to the topology of uniform convergence on compact sets, mapping $\mathbb{Q}$ into itself and satisfying $\operatorname{den}(f(p/q)) \le q^{2k+\varepsilon}$ , and deduce that such functions preserve Liouville numbers. In contrast, we prove a rigidity result about a $C^{2k+1}$ -function mapping $\mathbb{Q}$ into itself and satisfying $\operatorname{den}(f(p/q)) \ll q^k$ .

AI-generated audit

Audit summary

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.

Current report

Detailed mathematical audit

Generated August 15, 2026

01Statements3 reported findingsCorrect

The rigidity theorem, the dense uncountable bump-function construction, and the Liouville-preservation corollary are correct. Their printed proofs contain several substantive omissions and incorrect intermediate inferences, but each has a verified repair that preserves the stated hypotheses and conclusions.

Theorem 1.1Correct

Rigidity under the denominator bound

Pages 2 and 4–7 · Theorem 1.1 and Section 2.2 · arXiv:2607.24427v1

After cancelling the possible power of $x$ common to the Padé numerator and denominator, the rational approximant has degree at most $k$ and differs from the Taylor polynomial by one order more than the combined rational-denominator exponent. This still forces eventual zeros on every reciprocal sequence. The corrected divided-difference estimate then propagates equality across the connected interval, and a coprime representation excludes every genuine pole by continuity. These repairs verify the theorem with all of its stated derivative and denominator hypotheses.

Full paper, version 1 ↗

Theorem 1.2Correct

Dense uncountable family with controlled rational denominators

Pages 3 and 7–9 · Theorem 1.2 and Section 3.1 · arXiv:2607.24427v1

Order the rationals in $[0,1)$ by nondecreasing reduced denominator and choose all translated coefficients stage by stage. Rational separation makes every later bump vanish at each value already fixed, while every earlier contribution is fixed before the current coefficient is selected. A uniform convergent-denominator tail estimate establishes $C^k$ convergence; multiplying the normalization constant by a harmless fixed factor corrects the floor estimate. The support exclusion for $C(2)$ and the denominator upper bound are unchanged.

Full paper, version 1 ↗

Corollary 1.3Correct

Dense uncountable family preserving all Liouville numbers

Pages 3 and 10 · Corollary 1.3 and Section 3.3 · arXiv:2607.24427v1

At a rational of denominator $q$ , write $N=C\lfloor q^t\rfloor$ . The available interval of output values contains two consecutive multiples of $1/N$ ; at least one has reduced denominator at least $\sqrt N$ . Selecting it forces denominators to infinity, while the independent choices at the integer stages retain uncountably many functions. Lemma 3.1 then gives $f(\mathcal L)\subseteq\mathcal L$ . Finally, although the manuscript's uniqueness claim for algebraic functions is too strong, only countably many global algebraic functions can map every rational to a rational, so deleting them still leaves an uncountable family.

Full paper, version 1 ↗

02Proofs9 reported findingsContains incorrect or incomplete proofs

The main conclusions admit verified repairs, but the printed arguments contain false intermediate assertions in the Padé step and divided-difference estimate, omit the simultaneous-support and uniform-convergence checks in the bump construction, and assert denominator growth and algebraic uniqueness without valid proofs. Two literal notation defects are reported in yellow.

Theorem 1.1, Padé denominator at the originIncorrect as written · verified repair

The constructed denominator may vanish at zero

Pages 5–6 · Equation (3), properties (ii)–(iii), and the first eventual-zero argument · arXiv:2607.24427v1

The homogeneous system does not ensure $Q_k(0)\neq0$ . If $r=\operatorname{ord}_0Q_k>0$ , then $K=S/Q_k$ need not be bounded near zero and $g=f-\widetilde R_k$ need not have all derivatives through order $2k$ equal to zero. For example, when $k=1$ and $F_2(x)=x^2$ , the displayed system forces $Q_1$ to be a multiple of $x$ . The repair is to write $Q_k=x^rQ_0$ and, from $Q_kF_{2k}=P_k+x^{2k+1}S$ , also $P_k=x^rP_0$ . Then $\deg P_0,\deg Q_0\leq k-r$ , $R_k-F_{2k}=O(x^{2k+1-r})$ , and $\operatorname{den}(f(p/q)-R_k(p/q))\ll q^{2k-r}$ . The extra one power of vanishing still proves $g(n/q)=0$ for all sufficiently large $q$ . Repair classification: Verified repair.

Full paper, version 1 ↗

Theorem 1.1, divided-difference lower boundIncorrect as written · verified repair

The displayed estimate drops the factor $n^{2k+1}$

Page 6 · Displays following Equations (5)–(6) · arXiv:2607.24427v1

The exact formula contains $n^{2k+1}$ in the denominator, but the next lower bound writes $M_n^{2k+1}/M_n^{2k}=M_n$ after mentioning only $\binom{M_n+2k+1}{M_n}\geq1$ and $M_n\gg n$ . That inference does not follow from the stated lower bound. It is repaired by using $\binom{M_n+2k+1}{M_n}\geq\frac{M_n^{2k+1}}{(2k+1)!}.$ Together with $|g(n/M_n)|\gg M_n^{-2k}$ and $M_n\gg n$ , the exact formula gives $|g^{(2k+1)}(\xi_n)|\gg M_n$ , which is the required contradiction. Repair classification: Verified repair.

Full paper, version 1 ↗

Theorem 1.1, continuation across roots of $Q_k$Incorrect as written · verified repair

A root of the displayed denominator need not be a pole

Pages 6–7 · Final continuation argument · arXiv:2607.24427v1

The proof identifies every nonzero root of $Q_k$ with a pole of $P_k/Q_k$ , but $P_k$ and $Q_k$ were not shown to be coprime, so such a root can be removable. Cancel all common factors before defining the connected component $U$ . Its boundary points inside $\Omega$ are then genuine poles of the reduced rational function, and continuity of $f$ excludes them exactly as intended. The reciprocal-sequence argument also applies with negative numerators if a hypothetical nonzero interval lies to the left of the origin. Repair classification: Verified repair.

Full paper, version 1 ↗

Theorem 1.2, coefficient inductionIncomplete as written · verified repair

The evaluation omits other translates and does not justify future noninterference

Pages 7–8 · Definition of $f$ and the induction choosing $\lambda_{m,n}$ · arXiv:2607.24427v1

At $x=m+p_n/q_n$ , the displayed equality includes only bumps with the same translate $m$ . A bump from another translate can be nonzero: for instance, at $x=m+5/6$ , the integer-centered bump $\psi_{m+1,0}$ is active. The formula also does not show that later choices leave the selected value unchanged. Order the enumeration by nondecreasing $q_n$ , choose all integer coefficients first, and then proceed stagewise across $m$ . Earlier cross-translate terms are then fixed and can be included in the base value. If a later center $s$ has denominator $q_j\geq q_n$ , rational separation gives $|x-s|\geq\frac1{q_nq_j}\geq\frac1{q_j^2}>\frac1{4q_j^2},$ so its bump vanishes at $x$ . Thus every rational value remains fixed and retains its denominator bound. Repair classification: Verified repair.

Full paper, version 1 ↗

Theorem 1.2, $C^k$ convergence and distance estimateIncorrect as written · verified repair

Pointwise summability is not enough, and the floor inequality has the wrong direction

Page 8 · Equations (8)–(9) and the following distance estimate · arXiv:2607.24427v1

The paper infers $\widetilde f\in C^k$ from a pointwise convergent majorant without proving uniform convergence of the differentiated tails. In addition, $\lfloor q^t\rfloor\leq q^t$ , so the printed replacement of $q^{2j}/\lfloor q^t\rfloor$ by $q^{-(t-2j)}$ is not valid. For $q\geq2$ , one has $\lfloor q^t\rfloor\geq q^t/2$ . If the active denominator tail begins at $Q$ , the convergent recurrence gives $q_{i+s}\geq F_{s+1}q_i$ , hence the $j$ th derivative tail is uniformly bounded by a constant times $Q^{2j-t}\sum_{s\geq0}F_{s+1}^{\,2j-t},$ which tends to zero for $j\leq k$ because $t>2k$ . Taking $C$ strictly larger than $2(k+1)MA_t/\varepsilon$ then proves both termwise $C^k$ convergence and $d(f,g)<\varepsilon$ . Repair classification: Verified repair.

Full paper, version 1 ↗

Corollary 1.3, denominator growthIncomplete as written · verified repair

The required lower-denominator selection is absent

Page 10 · Sentence asserting $\operatorname{den}(f(p/q))\to\infty$ · arXiv:2607.24427v1

Theorem 1.2 as printed proves only an upper bound and does not itself imply that image denominators tend to infinity. At the stage for denominator $q$ , put $N=C\lfloor q^t\rfloor$ . Varying the coefficient makes the output range over an interval of length $2/N$ , which contains consecutive grid values $a/N$ and $(a+1)/N$ . The integers $\gcd(a,N)$ and $\gcd(a+1,N)$ are coprime and their product divides $N$ , so at least one of these two rationals has reduced denominator at least $\sqrt N$ . Selecting that value forces uniform denominator growth as $q\to\infty$ . Uncountability is retained by the two independent choices at every integer stage, which are irrelevant to the $q\to\infty$ condition. Repair classification: Verified repair.

Full paper, version 1 ↗

Corollary 1.3, algebraic exceptional functionsIncorrect as written · verified repair

Local equality does not force global equality for finite-smooth algebraic branches

Page 10 · Final identity-principle argument · arXiv:2607.24427v1

A global $C^k$ algebraic function can switch branches at a singular point. For example, the two $C^k$ functions obtained from the branches $\pm x^{k+1}\sqrt{1+x^2}$ can be chosen to coincide on $x>0$ and differ on $x<0$ ; both satisfy $y^2=x^{2k+2}(1+x^2)$ . Thus equality on one analytic interval does not prove that two such functions are globally identical. The corollary needs only countably many exceptions. The graph of an algebraic function is contained in a finite union of irreducible plane curves. If the function maps $\mathbb Q$ into $\mathbb Q$ , every component used over a nonempty interval contains infinitely many rational graph points; these are Zariski dense in that component, so Galois invariance shows that the component is defined over $\mathbb Q$ . There are only countably many finite unions of such rational curves. For each fixed union, the branch and pairwise-intersection abscissae form a finite set, and only finitely many global $C^k$ branch selections are possible. Thus the algebraic members form a countable set; removing them from the constructed uncountable family still proves the corollary. Repair classification: Verified repair.

Full paper, version 1 ↗

Divided-difference point countTypo

The number of interpolation points is off by one

Page 6 · First paragraph, immediately before Equation (4) · arXiv:2607.24427v1

The list $\alpha_0,\ldots,\alpha_{2k+1}$ contains $2k+2$ distinct points, not $2k+1$ . Replace the printed count by $2k+2$ . Equation (4) and every subsequent product already use the complete list, so the correction is mechanical and harmless.

Full paper, version 1 ↗

Definition of $\mathcal L_\Psi$Typo

The approximation exponent is unquantified

Page 9 · Lemma 3.1, definition of $\mathcal L_\Psi$ · arXiv:2607.24427v1

The displayed set uses an exponent $n$ without binding it. Replace the display by the sequential formulation used immediately in the proof: there exist distinct rationals $p_n/q_n$ with $q_n\to\infty$ and $0<|\xi-p_n/q_n|<(\Psi(q_n))^{-n}$ for every $n\geq1$ . This uniquely matches the subsequent argument and leaves Lemma 3.1 unchanged.

Full paper, version 1 ↗

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper: arXiv:2607.24427v1
Authors listed: Jean Lelis, Carlos Gustavo Moreira, Elaine Silva
Audit date: August 15, 2026

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You are a mathematical auditor. Evaluate only the correctness of the paper's central statements, the correctness and completeness of its proofs, and evidence-backed non-novelty. Do not make editorial comments or evaluate importance, exposition, style, or suitability for publication. Keep the report neutral, precise, and professional. PART 1 — CORRECTNESS OF THE MAIN STATEMENTS Identify the paper's central statements and assess each one separately using exactly one of these outcomes: - Correct: the statement has a correct, verifiable proof; or, despite a defect in the printed proof, you can give a fully verified repair or independent proof. - Incorrect: you can give strong formal evidence that the statement as written is false, preferably a counterexample or a complete contradiction. - Not able to verify: the supplied proof does not establish the statement, and you can neither supply a fully verified proof nor disprove it. 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For every defect, identify its smallest exact location, explain its downstream dependency, and classify any proposed repair as one of: - Verified repair: supply enough reasoning to verify that the repair works throughout every affected use. - Plausible repair only: describe the possible repair but do not treat it as established. - No repair supplied. Assign the overall proofs status by the following priority: 1. Contains incorrect or incomplete proofs, if at least one proof is Incorrect as written or Incomplete as written. 2. Contains unverified proofs, if there is no proved incorrectness or incompleteness but at least one material proof is Not able to verify. 3. Correct, if all audited proofs are Correct and complete, apart from any typographical or minor formal findings allowed by Part 4. PART 3 — NOVELTY OF THE MAIN STATEMENTS This is a one-sided, evidence-only check. Evaluate only central results. 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If the list is empty, the section may say only “No non-novelty findings.” PART 4 — TECHNICAL AND EVIDENTIARY RULES 1. Criticism must never be overstated. Call a statement incorrect, a proof incorrect or incomplete, or a result non-new only when strong formal evidence supports that exact judgment. Otherwise use Not able to verify where that outcome exists, or remain silent in the novelty section. 2. Review a specified paper version. Give exact page, theorem, equation, or proof locations; restate relevant quantifiers and hypotheses; and check domains, endpoints, parameter ranges, constants, and dependencies explicitly. Apply a defect only to the regimes and downstream results that actually depend on it. 3. Immediate-consequence closure: when deciding whether a conclusion is supported, treat the argument as including every immediate formal consequence of facts established earlier in the paper and available at that point. An immediate formal consequence is a local, mechanically checkable inference using only definitions already in force, elementary logic, direct substitution or renaming, weakening of an inequality or set inclusion, or a monotonicity or symmetry property explicit in a definition or already proved. 4. Such an inference must introduce no new lemma, estimate, construction, case analysis, limiting or compactness argument, external result, or new issue of uniformity or parameter dependence. Check the direction of monotonicity and the dependence of every threshold or constant explicitly. Do not exchange pointwise and uniform conclusions, change exceptional sets, interchange quantifiers or limits, or extend an endpoint under this rule unless the required fact was already established. 5. If the claimed conclusion follows after immediate-consequence closure, it is fully supported. 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P(delta)

is proved for every sufficiently small

delta>0

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P

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P(delta)

holds for every

delta>0

requires no correction; the omitted monotonicity step is exposition rather than an audit finding. 6. Immediate-consequence closure permits reconstruction of an omitted inference; it does not permit silent alteration of the printed domain, hypotheses, or quantifiers of an intermediate statement later invoked outside its stated scope. If formal validity requires changing the text of that statement or its stated range, report the appropriate finding even when the repair is easy. Thus a result stated only for

t>0

and subsequently applied at

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On CkC^kCk-functions mapping Q\mathbb{Q}Q into itself and Mahler's problem on Liouville numbers

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

Rigidity under the denominator bound

Dense uncountable family with controlled rational denominators

Dense uncountable family preserving all Liouville numbers

The constructed denominator may vanish at zero

The displayed estimate drops the factor n2k+1n^{2k+1}n2k+1

A root of the displayed denominator need not be a pole

The evaluation omits other translates and does not justify future noninterference

Pointwise summability is not enough, and the floor inequality has the wrong direction

The required lower-denominator selection is absent

Local equality does not force global equality for finite-smooth algebraic branches

The number of interpolation points is off by one

The approximation exponent is unquantified

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On $C^k$ -functions mapping $\mathbb{Q}$ into itself and Mahler's problem on Liouville numbers

The displayed estimate drops the factor $n^{2k+1}$