arXiv:2505.15964v3

Bad approximability, bounded ratios and Diophantine exponents

Antoine Marnat, Nikolay Moshchevitin, Johannes Schleischitz

math.NT11J13

Paper metadata

Original submission: May 21, 2025
Latest revision: July 21, 2026 · version 3
Claimed MSC: 11J13
arXiv categories: math.NT
Paper license: arXiv non-exclusive distribution license ↗
Comments: This version is corrected in accordance with referee's report, minor additional corrections delated to the definition of totally irrationality

Abstract

For a real $m\times n$ matrix $\pmbξ$ , we consider its sequence of best Diophantine approximation vectors $\pmb{x}_i \in \mathbb{Z}^n, \, i =1,2,3, ...$ , the sequences of its norms $X_i = \|\pmb{x}_i\|$ and the norms of remainders $L_i = \|\pmbξ\pmb{x}_i\|$ . It is known that, in the cases $m=1$ , bad approximability of $\pmbξ$ is equivalent to the boundedness of ratios $\frac{X_{i+1}}{X_i}$ , while for $n=1$ bad approximability of $\pmbξ$ is equivalent to the boundedness of ratios $\frac{L_i}{L_{i+1}}$ . Moreover, carefully constructed example show that in the cases $m=1$ and $n=1$ boundedness of ratios $\frac{L_i}{L_{i+1}}$ and $\frac{X_{i+1}}{X_i}$ respectively (the order of ratios changed), does not imply bad approximability of $\pmbξ$ . In the present paper, we study the impact of the boundedness of ratios on Diophantine properties of $\pmbξ$ , in particular, what restrictions it gives for Diophantine exponents $ω(\pmbξ)$ and $\hatω(\pmbξ)$ . One of our particular results deals with the case $m=n=2$ . We prove that for $2\times 2$ matrices $\pmbξ$ boundedness of both ratios $\frac{X_{i+1}}{X_i}, \frac{L_i}{L_{i+1}}$ implies inequality $\hatω(\pmbξ)\le \frac{4}{3}$ and that this result is optimal. Our methods combine parametric geometry of numbers as well as more classical tools.

AI-generated audit

Audit summary

Audited against arXiv v3

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

01Statements5 reported findingsContains unsupported statements

The norm-invariance theorem, the qualitative implication and counterexample results, the realizability theorems for ordinary and uniform exponents, and the stated upper bounds other than Theorem 2.8(2) are supported. No counterexample was found to Theorem 2.8(2), but its proof controls only infinitely many indices and therefore does not establish the required limsup bound. Theorem 2.10(2) also needs the local boundary correction $n\geq2$ (or the explicit hypothesis (2.6)).

Theorems 2.1–2.4Correct

Norm invariance, logical implications, counterexamples, and the three-dimensional nonsingularity result

Pages 4–5; proofs in Sections 4.1–4.2 and 5.1 · arXiv:2505.15964v3

Equivalence of norms and the bounded-distance description of successive-minima functions preserve properties (B) and (C). The implications in Theorem 2.2 and the $m+n=3$ nonsingularity argument follow with the stated hypotheses. For Theorem 2.3, the constructed connected templates have the required extremal-slope behavior; sparse excursions destroy exactly the selected property while their zero-density contribution leaves both contraction averages equal to $mn$ , giving the asserted full Hausdorff dimension through the cited variational principle.

Paper, version 3 ↗

Theorems 2.6 and 2.9Correct

Realizable ordinary and uniform exponents

Pages 5–6 and 23–33 · Sections 5.2–5.4 · arXiv:2505.15964v3

After the mechanically determined index correction recorded in Part 2, the families $\mathcal P_{v,w}$ tile the complete interval of admissible values of $\liminf P_1(q)/q$ while enforcing exactly the required combinations of (B), (C), and failure of (A). The families $\mathcal Q_v$ and $\mathcal R_u$ similarly cover the displayed intervals for $\limsup P_1(q)/q$ . Formula (3.12), connectedness, condition U, and the variational principle then yield totally irrational matrices with each claimed exponent and property combination. The edge case $n=m(m-1)$ is handled separately immediately before the indexed construction.

Paper, version 3 ↗

Theorem 2.8(1) and (3)Correct

Upper bounds for the ordinary exponent outside the case $m=2$

Pages 17 and 19 · Section 4.3.1 · arXiv:2505.15964v3

For $m=1$ , the covolume relation $L_{i-1}X_i\asymp1$ together with (C) gives $L_iX_i\asymp1$ and hence $\omega(\xi)\leq1$ . For $m\geq3$ , the quoted fixed-step growth $X_{k+3m+n-1}\geq2X_k$ makes $X_k$ exponentially large in $k$ , whereas (C) makes $-\log L_k$ at most linear in $k$ ; their ratio is therefore uniformly bounded, proving $\omega(\xi)<\infty$ .

Paper, version 3 ↗

Theorem 2.8(2)Not able to verify

The claimed bound $\omega(\xi)\leq2$ is not established

Pages 17–18 · proof of Theorem 2.8, Statement 2 · arXiv:2505.15964v3

In the nonplanar case, determinant estimates prove $L_{i+1}X_{i+1}^2\gg1$ (or the stronger $L_{i+1}^2X_{i+1}\gg1$ ) only at the infinitely many indices for which $x_{i-1},x_i,x_{i+1}$ are independent. But $\omega(\xi)=\limsup_{j\to\infty}\frac{-\log L_j}{\log X_j}$ requires control at every sufficiently large index, not merely along one infinite subsequence. Property (C) bounds successive ratios of the $L_j$ but supplies no bound on the intervening ratios $X_{j+1}/X_j$ , so immediate-consequence closure does not bridge those blocks. No later result repairs this: Lemma 4.2 assumes both (B) and (C), whereas this theorem assumes only (C). No counterexample to the statement was found, so the supported outcome is Not able to verify, not Incorrect.

Paper, version 3 ↗

Theorem 2.10Minor formal correction

Uniform-exponent upper bounds, with one boundary correction

Pages 6–7 and 19–22 · Theorem 2.10 and Section 4.3.2–4.3.3 · arXiv:2505.15964v3

Statement 1 is supported. Statement 2 is supported for $n\geq2$ , when total irrationality implies condition (2.6), and more generally whenever (2.6) is assumed. As printed, however, the theorem starts with $n\geq1$ and says that a totally irrational $\xi\in\mathbb R^{n\times2}$ therefore satisfies (2.6). For $n=1$ , the ambient space $\mathbb R^3$ is itself a three-dimensional rational subspace, so (2.6) is impossible; totally irrational badly approximable linear forms satisfy (B) and (C) and have $\widehat\omega=2$ , as Remark 2.11 itself notes. Replace $n\geq1$ by $n\geq2$ in Statement 2, or explicitly add (2.6). This removes only the identified boundary case, requires no new argument, and affects no later application.

Paper, version 3 ↗

02Proofs9 reported findingsContains incorrect or incomplete proofs

The proof of Theorem 2.8(2) has a substantive unresolved gap: an estimate on infinitely many independent triples does not control the limsup defining the ordinary exponent. The other central proof chains are verifiable after the localized typographical and boundary corrections listed below; those corrections do not lower the overall status independently.

Theorem 2.8(2)Incomplete as written

The independent-triple subsequence does not control all best approximants

Page 18 · two cases in the proof of Statement 2 · arXiv:2505.15964v3

The planar alternative is complete and even yields $\omega(\xi)=1$ . In the other alternative, the proof obtains a lower bound for $L_{i+1}$ only whenever three successive best-approximation vectors are independent, then concludes from the infinitude of those indices that $\omega(\xi)\leq2$ . A limsup upper bound instead requires a corresponding estimate throughout every intervening dependent block. Condition (C) controls $L_i/L_{i+1}$ , but without (B) it does not control the growth of $X_i$ across such a block. Repair classification: No repair supplied; a new block estimate under (C) alone, or another global argument, is required.

Paper, version 3 ↗

Theorem 2.10(2)Correct and complete

Pattern argument and the $4/3$ bound

Pages 20–22 and Appendix A · proof of Theorem 2.10(2) · arXiv:2505.15964v3

Under condition (2.6) and after the local corrections listed below, successive independent triples delimit a block lying in one two-dimensional subspace. Lemma 4.2 and (B) give $L_\nu\ll X_\nu^{-1}X_k^{1-\alpha}$ , while an independent-triple determinant gives $L_\nu\gg X_\nu^{-2}$ and the four-vector determinant gives $1\ll L_{\nu-1}L_{k-1}X_kX_{k+1}$ . Thus $X_k^{\alpha-1}\ll X_\nu\ll X_k^{3-2\alpha},$ forcing $\alpha\leq4/3$ . Since this holds for every $1<\alpha<\widehat\omega(\xi)$ , the claimed bound follows.

Paper, version 3 ↗

Theorem 2.8(2) · proportional-coordinate determinantTypo

The second error row uses the wrong coefficients

Page 18 · final determinant in the proportional case · arXiv:2505.15964v3

In the row beginning with $y_{l,i-1}$ , the subtracted coefficients are printed as $\xi_{j,1},\xi_{j,2}$ . They must be $\xi_{l,1},\xi_{l,2}$ in all three entries. The row label $l$ and the definition of $L_i$ determine this correction uniquely; with it both lower rows are $O(L)$ and the displayed determinant bound follows unchanged.

Section 4.3.3 · vector labelsTypo

Two local notation errors have unique corrections

Pages 20–21 · opening paragraph and Equation (4.7) · arXiv:2505.15964v3

The matrix $\xi$ is called $2\times n$ although the paper's convention and every displayed vector require $\xi\in\mathbb R^{n\times2}$ . Equation (4.7) then repeats $x_{k-1}$ in a purported list of four independent vectors; the immediately following rank-four matrix shows that the list must be $x_{\nu-1},\ x_{k-1},\ x_k,\ x_{k+1}.$ Both replacements are mechanically determined and the subsequent calculation already uses the corrected objects.

Section 5.2 · template parametersTypo

A sign omission and an exponent-index shift need correction

Pages 26–29 · Remark 5.4 and proofs of Theorem 2.6(1),(3) · arXiv:2505.15964v3

Remark 5.4 prints $\lambda_k\geq\log q_k$ , impossible because $\lambda_k\leq0$ ; the singularity condition is $-\lambda_kq_k\geq\log q_k$ (equivalently $\lambda_kq_k\leq-\log q_k$ ). Later, for $n>m(m-1)$ , the printed choice $m^{\ell_n-1}-m<n\leq m^{\ell_n}-m$ makes $v_{\ell_n}=m^{\ell_n}-m\geq n$ , contrary to the required $v<n$ . The uniquely consistent shift is $m^{\ell_n}-m<n\leq m^{\ell_n+1}-m.$ It makes every $v_\ell$ admissible and makes intervals (5.7)–(5.9) tile $[-n,0]$ ; the equality $n=m(m-1)$ was already handled in the preceding paragraph. Apply the same shift in Section 5.2.6.

Section 5.3 · interval notationTypo

The rising segment has the wrong right endpoint

Page 29 · first paragraph of Section 5.3 · arXiv:2505.15964v3

After choosing $q_k<p_k<q_{k+1}$ , the text says that $Q_1$ has positive or constant slope on $[p_k,q_k]$ , a reversed interval. The constructions immediately below determine the intended segment as $[p_k,q_{k+1}]$ . This correction changes no template or exponent calculation.

Appendix A · ellipse definitionTypo

The error threshold must be $L_l$

Page 34 · definition of $G_l$ in the proof of Lemma A.1 · arXiv:2505.15964v3

The ellipse is printed as $G_l=\pi\cap\{x:\lVert\xi x-y\rVert_2\leq L_{l+1}\},$ but the next lines require both $z_l,z_{l+1}\in G_l$ and explicitly use $z_l\in G_l$ . Since the errors decrease strictly, $z_l$ need not satisfy the printed threshold. Replace $L_{l+1}$ by $L_l$ . This is uniquely forced by those inclusions and makes the ensuing area and covolume argument valid without changing its bounds.

Appendix A · index rangeMinor formal correction

One endpoint is short by one index

Pages 33–34 · Case 2 in the proof of Lemma A.1 · arXiv:2505.15964v3

With $k_1=k-C_2$ , the hypothesis $k\geq\nu+2C_2$ yields only $k_1\geq\nu+C_2$ . Applying (A.5) with starting index $\nu+1$ to conclude $X_{k_1}\geq Z_{\nu+1}$ requires $k_1\geq\nu+1+C_2$ . Increase the Case 2 threshold to $k\geq\nu+2C_2+1$ and absorb the remaining single index into Case 1 (or replace $k_1$ by $k-C_2+1$ ). The finite-step comparison (A.4) already covers this local adjustment, so no later estimate changes.

Remaining central proofsCorrect and complete

Best-approximation and template proof chains

Sections 4.1–4.2 and 5.1–5.4 · arXiv:2505.15964v3

The bounded-distance minima lemma is applied with the correct norm-equivalence constants; the low-dimensional covolume arguments preserve total irrationality and all cases; and the template constructions satisfy the slope, ordering, summation, connectedness, and extremal-slope conditions needed by the variational principle. Sparse exceptional intervals contribute zero asymptotic density, while the exponent formulas use the correct liminf or limsup in each construction. No further nontrivial omitted obligation was identified.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper: arXiv:2505.15964v3
Authors listed: Antoine Marnat, Nikolay Moshchevitin, Johannes Schleischitz
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)

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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

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Bad approximability, bounded ratios and Diophantine exponents

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

Norm invariance, logical implications, counterexamples, and the three-dimensional nonsingularity result

Realizable ordinary and uniform exponents

Upper bounds for the ordinary exponent outside the case m=2m=2m=2

The claimed bound ω(ξ)≤2\omega(\xi)\leq2ω(ξ)≤2 is not established

Uniform-exponent upper bounds, with one boundary correction

The independent-triple subsequence does not control all best approximants

Pattern argument and the 4/34/34/3 bound

The second error row uses the wrong coefficients

Two local notation errors have unique corrections

A sign omission and an exponent-index shift need correction

The rising segment has the wrong right endpoint

The error threshold must be LlL_lLl​

One endpoint is short by one index

Best-approximation and template proof chains

Challenge an audit finding

Authenticate with ORCID, then require an exact arXiv match

Upper bounds for the ordinary exponent outside the case $m=2$

The claimed bound $\omega(\xi)\leq2$ is not established

Pattern argument and the $4/3$ bound

The error threshold must be $L_l$