arXiv:2508.18044v6

Diophantine approximation with sums of two squares II

Stephan Baier, Habibur Rahaman

math.NT11J2511J5411J7111E25

Paper metadata

Original submission: August 25, 2025
Latest revision: July 8, 2026 · version 6
Claimed MSC: 11J25, 11J54, 11J71, 11E25
arXiv categories: math.NT
Paper license: arXiv non-exclusive distribution license ↗
Comments: This is a reworked version. We have managed to establish the same result by simpler means, resulting in a reduction of pages from 17 to 15

Abstract

Recently, the authors showed that for every irrational number $α$ , there exist infinitely many positive integers $n$ represented by any given positive definite binary quadratic form $Q$ , satisfying $||αn||<n^{-(1/2-\varepsilon)}$ for any fixed $\varepsilon>0$ . We also provided a quantitative version with a lower bound when the exponent $1/2-\varepsilon$ is replaced by a smaller exponent $γ<3/7-\varepsilon$ . In this article, we establish a quantitative version for the exponent $1/2-\varepsilon$ , where we confine ourselves to the particular case of sums of two squares.

AI-generated audit

Audit summary

Audited against arXiv v6

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 asymptotic formula for the smoothed representation sum and the resulting quantitative Diophantine-approximation lower bound are correct. The lower-bound clause for the smoothed sum implicitly requires the two nonnegative test functions to be nonzero, a harmless boundary condition recorded below.

Theorem 3Correct

Asymptotic formula for the smoothed sum-of-two-squares count

Pages 2 and 5–14 · Theorem 3 and Section 3 · arXiv:2508.18044v6

The divisor convolution for $r_2(n)$ , the dyadic partition, and Poisson summation produce the stated main term. Möbius inversion gives the factors $\varphi(q)/q$ and $\prod_{p\mid q}(1-\chi_4(p)/p)$ . The complementary error estimates are $O(X^{4\varepsilon}L^2(X^{-1/2}+L^{-1}+X^{1/2}L^{-2}))$ ; with $L=q^\beta$ , $X=q^{1+\beta}$ , and $\beta>1/3$ , every term is $o(L^2/(\log\log q)^2)$ . This yields both the asymptotic and the displayed lower bound.

Corollary 1Correct

Quantitative approximation with sums of two squares

Pages 3 and 14–15 · Corollary 1 and Section 4 · arXiv:2508.18044v6

For $|\alpha-a/q|\leq24/q^2$ , a represented integer counted by the smoothed sum satisfies $\|\alpha n\|\ll L/q+X/q^2\asymp X^{-\gamma}$ when $\gamma=(1-\beta)/(1+\beta)$ . The standard maximal-order bound $r_2(n)\ll X^{c/\log\log X}$ converts the weighted representation count into the asserted number of distinct integers, and $L^2=X^{1-\gamma}$ gives the claimed exponent. Lemma 2.1 supplies infinitely many admissible approximants.

Minor boundary correctionMinor formal correction

The lower bound assumes nonzero test functions

Page 2 · statement of Theorem 3 · arXiv:2508.18044v6

As written, the test functions are arbitrary nonnegative compactly supported functions, so they may vanish identically or be rescaled by an arbitrarily small constant; $\Phi\equiv0$ , for example, gives $S=0$ and contradicts an absolute positive lower bound. State that fixed nonzero tests are used and write $S\gg_{\Phi,w}L^2/(\log\log q)^2$ , or retain the factor $\widehat\Phi(0)\widehat w(0)$ . This is a local normalization correction and does not affect Corollary 1, where fixed positive bumps may be chosen.

02Proofs4 reported findingsCorrect

The reworked version 6 proof is correct and complete. The small- and large-divisor ranges, the zero-frequency main term, and the off-frequency divisor estimates fit together with the required uniform margins for every fixed $1/3<\beta<1$ .

Lemmas 3.1–3.3Correct and complete

Main term and both complementary error ranges

Pages 7–14 · Lemmas 3.1–3.3 and proof of Theorem 3 · arXiv:2508.18044v6

The coprimality conditions justify every modular inverse used in the Poisson transformations. In the small-divisor range, reciprocity converts the remaining phase to a divisor condition $d\mid(uq-ghc)$ ; separating the zero difference from the nonzero differences gives the three terms in Lemma 3.2. Interchanging $m$ and $d$ and splitting odd $d$ modulo $4$ gives the same estimate in the large-divisor range, with the zero frequencies cancelling. Summation over dyadic scales produces exactly the error quoted in Lemmas 3.2 and 3.3.

Minor formal correctionMinor formal correction

Two dyadic support ranges are too short by a constant factor

Pages 5 and 11–13 · Equations (3.3)–(3.5) and Section 3.5 · arXiv:2508.18044v6

Because $\Omega(d/D)$ is supported on $[1/2,2]$ and $w(md/X)$ on $[1,2]$ , the exact ranges are $D\leq4X$ and $X/(2D)\leq m\leq4X/D$ , rather than $D\leq2X$ and $m\leq2X/D$ . With these replacements, $Dm/X$ remains in the fixed interval $[1/2,4]$ , so the Fourier-decay constants for $W_m$ remain uniform and the identical counting estimate proves Lemma 3.3 after summing the one additional dyadic scale. No exponent or conclusion changes.

Minor parameter correctionMinor formal correction

The auxiliary $\varepsilon$ must satisfy all three error margins

Page 13 · Section 3.6 · arXiv:2508.18044v6

The condition $4\varepsilon<\beta-1/3$ alone is not sufficient for every $1/3<\beta<1$ to make all three normalized error terms tend to zero. It is enough to choose $4\varepsilon<\min\left\{\frac12,\frac{\beta}{1+\beta},\frac{3\beta-1}{2(1+\beta)}\right\}.$ Since $\varepsilon$ is a free auxiliary parameter and the minimum is positive throughout the stated range, this is a local correction with no effect on Theorem 3.

TypoTypo

Four isolated notation slips

Pages 3–4, 12, and 14 · Notations, Lemma 2.3, Section 3.5, and proof of Corollary 1 · arXiv:2508.18044v6

The little- $o$ definition reverses the quotient and should read $f(x)/g(x)\to0$ . Under the declared Fourier transform, Lemma 2.3 should contain $\widehat\Phi(L(n/q-\omega))$ rather than a plus sign; all later uses take absolute values, so they are unchanged. The minus branch in Section 3.5 writes $k_1$ after introducing $k_2$ . Finally, the approximants in Section 4 should be $(a,q)\in\mathbb Z\times\mathbb N$ , not $\mathbb N\times\mathbb Z$ . Each correction is uniquely determined by the surrounding formulas.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

Detailed audit reportFull reasoning, manuscript locations, and references.

Open report PDF ↗

Author response

Challenge an audit finding

Local workflow preview

A listed author may submit formal evidence that an audit is inaccurate. The response would be considered in a fresh AI re-evaluation; it would not edit the audit automatically.

Paper: arXiv:2508.18044v6
Authors listed: Stephan Baier, Habibur Rahaman
Audit date: August 15, 2026

01Establish identityMatch an authenticated scholarly identity to this paper.
02Submit evidenceIdentify the finding and give a formal mathematical response.
03Re-evaluateA separate agent checks the response and records a disposition.

Email fallback for papers without a linked ORCID

A production fallback could send a one-time link only when the submitted address matches an independently maintained author-contact allowlist for this paper. MathAudit must return the same message for every address so the form cannot reveal which contacts are on that list.

Structured response preview

This form remains unavailable until production identity verification succeeds. Nothing entered here is submitted.

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

This panel never establishes authorship in the local prototype. A production result should be described narrowly as an authenticated ORCID match or control of a separately allowlisted author-contact mailbox.

Audit controls

Status: Report ready
Source: arXiv v6
Prompt: Standard prompt · locked

The report is available below and in the public summary.

Review prompt

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.

This panel is visibly marked as owner-only for the concept prototype. Authentication is not enabled locally.

Diophantine approximation with sums of two squares II

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

Asymptotic formula for the smoothed sum-of-two-squares count

Quantitative approximation with sums of two squares

The lower bound assumes nonzero test functions

Main term and both complementary error ranges

Two dyadic support ranges are too short by a constant factor

The auxiliary ε\varepsilonε must satisfy all three error margins

Four isolated notation slips

Challenge an audit finding

Authenticate with ORCID, then require an exact arXiv match

The auxiliary $\varepsilon$ must satisfy all three error margins