arXiv:2502.00731v2

Diophantine approximation and the subspace theorem

Shivani Goel, Rashi Lunia, Anwesh Ray

math.NT11J87

Paper metadata

Original submission: February 2, 2025
Latest revision: July 21, 2026 · version 2
Claimed MSC: 11J87
arXiv categories: math.NT
Paper license: CC BY 4.0 ↗
Comments: Version 2: 52 pages; accepted for publication in La Matematica (official journal of the Association for Women in Mathematics)
Journal reference: La Matematica, Volume 5, article number 52 (2026)
DOI: 10.1007/s44007-026-00232-x ↗

Abstract

Diophantine approximation explores how well irrational numbers can be approximated by rationals, with foundational results by Dirichlet, Hurwitz, and Liouville culminating in Roth's theorem. Schmidt's subspace theorem extends Roth's results to higher dimensions, with profound implications to Diophantine equations and transcendence theory. This article provides a self-contained and accessible exposition of Roth's theorem and Schlickewei's refinement of the subspace theorem, with an emphasis on proofs. The arguments presented are classical and approachable for readers with a background in algebraic number theory, serving as a streamlined, yet condensed reference for these fundamental results.

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Audit summary

Audited against arXiv v2

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

01Statements2 reported findingsCorrect

The article's central statements—Roth's theorem over number fields and Schlickewei's refinement of the subspace theorem—are correct. The proof text contains several genuine defects, but each one identified below has a direct verified repair and no central theorem statement needs to be changed.

Theorems 3.2–3.3Correct

The two number-field forms of Roth's theorem

Pages 15–25 · Theorems 3.2–3.3 and Section 3.6 · arXiv:2502.00731v2

Both formulations are the standard number-field Roth theorem. The converse implication in Proposition 3.4 and the integral reduction omitted in Section 3.6 are defective as printed, but the finite-subset discretization and denominator-clearing repairs stated under Proofs establish the missing implications without changing either theorem. The auxiliary-polynomial and index arguments then give the claimed finiteness result.

Theorems 4.3–4.4Correct

Affine and projective subspace theorems

Pages 27–52 · Theorems 4.3–4.4 and Sections 4–5 · arXiv:2502.00731v2

The affine and projective conclusions are Schlickewei's subspace theorem in the stated adelic setting. The proof's non-vanishing Lemma 5.1 chooses a coefficient that need not be nonzero, but choosing any nonzero coefficient gives the same induction with all degree bounds preserved. With that local repair, the auxiliary-polynomial, index, and rank argument yields the finite union of proper subspaces claimed.

02Proofs4 reported findingsContains incorrect or incomplete proofs

Three printed arguments require correction: the finite discretization in the equivalence of Roth formulations, the reduction from an arbitrary algebraic target to an algebraic integer, and the induction in the multivariable non-vanishing lemma. Each defect has a verified repair, so the theorem statements remain correct. One parameter is also a uniquely identifiable typo.

Proposition 3.4Incorrect as written · verified repair

The printed discretization does not imply the required local exponents

Pages 15–16 · Proposition 3.4 · arXiv:2502.00731v2

The proof chooses integers $b_v\leq2s\lambda_v$ and sets $\zeta(v)=b_v/s$ . This gives only $\zeta(v)\leq2\lambda_v$ ; it does not imply $(2+\delta)\zeta(v)\leq(2+\varepsilon)\lambda_v$ , which is needed to pass from the product inequality to every local inequality. It also does not handle places at which $\min(1,|\beta-\alpha|_v)=1$ but $|\beta-\alpha|_v\not<1$ . Repair classification: Verified repair. First split solutions among the finitely many subsets $T\subseteq S$ on which $|\beta-\alpha|_v<1$ . For a fixed $T$ , choose $0<\delta<\varepsilon$ , then choose a sufficiently fine finite rational grid of weights $\zeta$ with $\sum_{v\in T}\zeta(v)=1$ and $\zeta(v)\leq(1+\eta)\lambda_v$ , where $(2+\delta)(1+\eta)\leq2+\varepsilon$ . Apply Theorem 3.3 to $T$ and each grid point. This proves the converse with finitely many cases.

Proof of Theorem 3.3Incomplete as written · verified repair

The reduction to an algebraic-integer target is omitted

Pages 17 and 24–25 · Lemma 3.6 and Section 3.6 · arXiv:2502.00731v2

Lemma 3.6 assumes that $\alpha$ is an algebraic integer, while Theorem 3.3 permits every algebraic $\alpha$ and Section 3.6 invokes the lemma without making the reduction. Repair classification: Verified repair. Fix a nonzero integer $a$ for which $a\alpha$ is integral and replace $\beta$ by $a\beta$ . Multiplication by the fixed $a$ changes heights and the finitely many local factors by constants depending only on $a$ , which are absorbed after discarding a bounded-height exceptional set. The integral case therefore implies the stated general case.

Lemma 5.1Incorrect as written · verified repair

The induction may select the zero coefficient

Pages 42–43 · Lemma 5.1 · arXiv:2502.00731v2

Writing $f=A_0+A_1x_N+\cdots+A_dx_N^d$ , the proof sets $g=A_0$ and applies induction to it, although $A_0$ may be zero; for example, $f=x_N$ . Repair classification: Verified repair. Choose any index $k$ with $A_k\neq0$ and apply the induction hypothesis to $A_k$ . After differentiating and specializing the first $N-1$ variables, the resulting univariate polynomial remains nonzero because its $x_N^k$ coefficient is nonzero. The one-variable case then supplies $z_N$ and $i_N$ with exactly the printed bounds. This repairs every downstream use in the proof of Theorem 4.14.

Lemma 3.8 and Section 3.6Typo

The height exponent uses the target instead of the approximation parameter

Pages 19 and 24 · Lemma 3.8 and the choice of $\beta_1$ · arXiv:2502.00731v2

The two displayed height thresholds contain $8/\alpha+4/11$ . Replace $\alpha$ there by $\delta$ , giving $8/\delta+4/11$ . The end of the proof of Lemma 3.8 derives precisely the exponent $8(1+\varepsilon)/\delta<8/\delta+4/11$ , and Section 3.6 copies the same threshold. The correction is unique and does not alter the argument.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper: arXiv:2502.00731v2
Authors listed: Shivani Goel, Rashi Lunia, Anwesh Ray
Audit date: August 15, 2026

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Audit finding challengedWhy the evaluation is inaccurateFormal evidence or repairVerifiable sources, if anyI am submitting evidence, not instructions for the reviewing agent.

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Source: arXiv v2
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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. For every adverse or unverified outcome, restate the exact claim and hypotheses, identify its location in the reviewed version, and give the decisive reasoning. Do not infer that a theorem is false merely because its printed proof is defective. Assign the overall statements status by the following priority: 1. Contains wrong statements, if at least one central statement is Incorrect. 2. Contains unsupported statements, if no central statement is Incorrect but at least one is Not able to verify. 3. Correct, if every central statement is Correct, apart from any typographical or minor formal findings allowed by Part 4. PART 2 — CORRECTNESS AND COMPLETENESS OF THE PROOFS Audit the proofs of the central statements and every material internal or cited input on which they depend. For each proof or proof segment, use the narrowest supported outcome: - Correct and complete: every substantive step is verifiable and no case or crucial nontrivial step is missing. - Incorrect as written: a displayed calculation, inference, lemma, or cited application is formally false or inapplicable. - Incomplete as written: a necessary substantive step or case is missing, even though no explicit contradiction has been proved. - Not able to verify: the available paper and sources are insufficient to decide whether the step is valid. Do not flag omitted routine details, compressed arguments, or “analogously” steps when the omitted reasoning is recoverable and correct. 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. Report a result only when a verifiable primary source, publicly available before the earliest public version of the paper that contains the result, proves the same statement up to a change of notation, or when the result follows directly and immediately from such an earlier theorem. For each non-novelty finding, provide the earlier source, its public date, the exact theorem or proposition, and an explicit hypothesis-and-conclusion mapping. A later source cannot establish non-novelty. Do not flag a result merely because an easier proof exists or because it follows through a nontrivial argument. Do not flag a result that the paper itself explicitly presents as known, a survey item, or an alternative proof. Mention only statements that meet this non-novelty threshold. Say nothing about statements for which no such evidence was found: do not certify novelty, list unsuccessful searches, or add statement-level “no prior source found” entries. 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. Do not report a gap, unsupported statement, minor formal correction, or recommendation to add an explanatory sentence, and do not mention the suppressed item in the report. This suppression does not apply to a genuine typographical error as defined below, because a typo is a literal notation defect that still requires a textual correction. For example, if

P(delta)

is proved for every sufficiently small

delta>0

and

P

is immediately monotone in the required direction, then the claim that

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

t=0

remains a formal scope mismatch unless the endpoint case was separately established before the application. 7. Apply this mandatory gate before recording any adverse, unverified, typographical, or minor formal finding: (a) identify the exact assertion or inference alleged to be defective; (b) reconstruct the strongest conclusion following from the preceding argument after all admissible immediate formal consequences are added; (c) identify the specific defect that remains — a formally demonstrated falsehood, a concrete nontrivial unresolved proof obligation, or a genuine domain, hypothesis, quantifier, or notation mismatch requiring a textual correction; and (d) state the evidence and minimal repair. If no specific defect remains, or if the only possible addition would be explanatory, suppress the finding entirely. Use Not able to verify only when a precise, nontrivial unresolved obligation can be identified. 8. A typographical error is an incorrect symbol, index, exponent, sign, variable name, or notation mismatch whose unique intended correction is mechanically determined from the surrounding text and does not alter the mathematical argument. Report it in yellow as Typo, with the printed text, correction, exact location, and reason it is harmless. It does not lower an overall status. 9. A minor formal correction is a boundary, endpoint, or displayed parameter-range defect that can be repaired consistently in every affected statement by a local change, without a new argument, without invalidating any related result or application, and without creating another substantive defect. Report it in yellow as Minor formal correction, with the same details. It does not lower an overall status. 10. Do not use either minor category if more than one materially different repair is possible; if the repair changes the substance or scope of a central result; if an excluded case is used later; if the correction fails to propagate consistently; or if a new mathematical idea is required. In such cases use the strongest substantive classification actually supported by evidence. 11. Write every mathematical expression as valid, compilable LaTeX. Use

...

for inline mathematics and

...

for displayed mathematics, with balanced delimiters and braces. Do not leave raw LaTeX commands outside math delimiters. 12. Separate theorem truth from the correctness of the printed proof. A theorem can be Correct in Part 1 because of a fully verified independent proof while its printed proof is adverse in Part 2. Conversely, a defective proof alone never licenses an Incorrect theorem classification. Organize the report under Parts 1–3 and apply Part 4 throughout. Do not add editorial commentary, scores, rankings, recommendations, or a list of suppressed observations.

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Diophantine approximation and the subspace theorem

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

The two number-field forms of Roth's theorem

Affine and projective subspace theorems

The printed discretization does not imply the required local exponents

The reduction to an algebraic-integer target is omitted

The induction may select the zero coefficient

The height exponent uses the target instead of the approximation parameter

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