arXiv:2606.16699v1

Quantitative Oppenheim Conjecture for Random Quadratic Forms and Optimal Variance Bounds in Function Fields

Jiyoung Han, Noy Soffer Aranov

math.NTmath.DSmath.PR11H0611K3811J61

Paper metadata

Original submission: June 15, 2026
Claimed MSC: 11H06, 11K38, 11J61
arXiv categories: math.NT, math.DS, math.PR
Paper license: CC BY 4.0 ↗

Abstract

We prove a quantitative version of Oppenheim's conjecture in the function field setting. In order to do so, we compute the higher moments of the Siegel transform. In particular, we find an optimal bound on the variance of the number of lattice points in a set. Moreover, we compute the exact variance of the number of lattice points in a ball, which is of independent interest.

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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 findingsContains unsupported statements

The Siegel mean formula, variance bound, exact ball variance after a verified repair, shell-volume formula for ball targets, and bounded moments of the Margulis function are supported. Theorem 1.1 is not able to be verified for an arbitrary bounded measurable target because the required quantitative volume estimate is proved only for a ball.

Theorem 1.1Not able to verify

The arbitrary-measurable-target scope is not established

Page 3 · Theorem 1.1; pages 17–22 · Theorems 4.3, 4.9, and proof of Theorem 1.1 · arXiv:2606.16699v1

Theorem 1.1 allows every bounded measurable $I\subset K_\infty$ . The volume theorem used in its proof, Theorem 4.3, assumes that $I=a+x^b\mathcal O$ is a one-dimensional ball. Nevertheless, estimate (4.13) invokes Theorem 4.3 for the arbitrary measurable set appearing in Theorem 4.9, and the final proof again invokes it in that scope. Approximation by finite unions of balls gives qualitative measure approximation, but for a completely arbitrary measurable set it does not automatically give the exponential accuracy needed for the stated error $O(q^{\delta t(n-2)})$ . No boundary regularity or quantitative approximation argument is supplied. The result for a ball, and plausibly for target classes with suitable quantitative approximation, is supported; the full measurable-set statement is not.

Full paper, version 1 ↗

Proposition 1.7Correct

The exact variance formula survives a correction to its counting step

Pages 4 and 16–17 · Proposition 1.7 and proof · arXiv:2606.16699v1

The per-polynomial count printed after (3.28) is false, but the aggregate count needed in the variance sum is correct. Writing $a=Q+b$ and summing first over nonzero $b$ with $\deg b\leq d-t$ , the number of monic $Q$ of degree $d$ coprime to a fixed $b$ of degree $e$ is $q^{d-e}\varphi(b)$ . Summing over $b$ gives $(q-1)q^{2d-t}$ , exactly the aggregate value used in the subsequent geometric series. Thus the displayed variance formula follows after this verified reordering repair.

Full paper, version 1 ↗

Theorems 1.1 and 1.2Minor formal correction

The scale and leading-constant notation need local corrections

Pages 3 and 21–22 · Theorems 1.1–1.2 and their proofs · arXiv:2606.16699v1

Norms on $K_\infty^n$ take values in $q^{\mathbb Z}$ , and the shell proof sums over integer exponents, so the quantifier on $t$ should specify $t\in\mathbb Z$ rather than all real $t>t_{Q,I}$ . Also, the shell-volume coefficient is $c_Q=J_Qq^{n-2}/(q^{n-2}-1)$ , while the Siegel mean contributes an additional factor $q^n$ to the lattice-count coefficient in (4.14). Calling both coefficients $C_Q$ creates a normalization mismatch. Distinguishing $c_Q$ from $C_Q=q^n c_Q$ and restricting the scale to integer exponents are unique local corrections; neither changes the substance of the asymptotic statement.

Full paper, version 1 ↗

02Proofs2 reported findingsContains incorrect or incomplete proofs

The proof of the quantitative Oppenheim theorem uses the ball-volume theorem outside its hypotheses. The proof of the exact variance formula also contains a false individual coprimality count, although a direct aggregate recount repairs that formula.

Proof of Theorems 4.9 and 1.1Incomplete as written

The Borel–Cantelli estimate applies a ball theorem to an arbitrary measurable set

Pages 17–22 · Theorem 4.3, estimate (4.13), and proof of Theorem 1.1 · arXiv:2606.16699v1

Theorem 4.3 proves the required volume law only when $I$ is a ball. In (4.13), the proof cites that theorem to obtain $\operatorname{Vol}(A_{h_i,I,t})\asymp q^{t(n-2)}$ for a merely measurable $I$ , and the final proof requires the sharper leading-term formula in the same unsupported scope. Repair classification: No repair supplied for arbitrary measurable targets. Restricting Theorems 4.9 and 1.1 to ball targets is a verified scope repair; extending them further requires an explicit target-approximation theorem with a rate compatible with the claimed error.

Full paper, version 1 ↗

Proof of Proposition 1.7Incorrect as written

The individual count after (3.28) is false

Pages 16–17 · equation (3.28) and the following inclusion–exclusion calculation · arXiv:2606.16699v1

The manuscript claims that for each monic $Q$ of degree $d\geq t$ , the count in (3.28) equals $\varphi(Q)/q^{t-1}$ . If $Q$ is irreducible and $d=t\geq2$ , the admissible polynomials are $a=Q+c$ with $c\in\mathbb F_q^\times$ , so the count is $q-1$ , whereas the printed expression is $(q^d-1)/q^{d-1}$ and is not even an integer. The inclusion–exclusion step improperly assigns fractional counts when a divisor degree exceeds $d-t$ . Repair classification: Verified repair. Sum first over $b=a-Q$ as described in the statement finding; the aggregate count is $(q-1)q^{2d-t}$ and the remainder of the variance computation is unchanged.

Full paper, version 1 ↗

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

Detailed audit reportFull reasoning, manuscript locations, and references.

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Paper: arXiv:2606.16699v1
Authors listed: Jiyoung Han, Noy Soffer Aranov
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. 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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Quantitative Oppenheim Conjecture for Random Quadratic Forms and Optimal Variance Bounds in Function Fields

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

The arbitrary-measurable-target scope is not established

The exact variance formula survives a correction to its counting step

The scale and leading-constant notation need local corrections

The Borel–Cantelli estimate applies a ball theorem to an arbitrary measurable set

The individual count after (3.28) is false

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