arXiv:2607.17021v1

Bounded trajectories of quasi-rays on homogeneous spaces and Diophantine approximation with weight functions

Dmitry Kleinbock, Vasiliy Neckrasov

math.DSmath.NT37A1737A2511J1317B45

Paper metadata

Original submission: July 19, 2026
Claimed MSC: 37A17, 37A25, 11J13, 17B45
arXiv categories: math.DS, math.NT
Paper license: arXiv non-exclusive distribution license ↗
Comments: 32 pages

Abstract

Let $G$ be a connected semisimple real Lie group, $Γ$ an irreducible lattice in $G$ and $X = G/Γ$ . Let $F = \{g_t: t\ge 0\}$ be a non-quasiunipotent one-parameter subsemigroup of $G$ . Then it is known that the set of points in $X$ with bounded $F$ -trajectories has full Hausdorff dimension. In addition, if $U$ is the expanding horospherical subgroup relative to $g_1$ , then for any $x \in X$ the set of points $u \in U$ such that the $F$ -trajectory of $ux$ is bounded has full Hausdorff dimension. In this paper we take $U$ to be a horospherical subgroup of $G$ and apply Shi's equidistribution theorem for elements of the expanding cone with respect to $U$ to describe a class of subsets $F$ in $G$ , not presupposing the group structure, for which the above full Hausdorff dimension statements also hold. As an application, we prove that the set of badly approximable matrices in the set-up of Diophantine approximations with quasimultiplicative weight functions has full Hausdorff dimension.

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

01Statements4 reported findingsCorrect

The central results checked are correct under their stated hypotheses and cited inputs. The only detected defect is a mechanical index typo in the power-weight example; it is reported in yellow and does not lower the substantive status.

Theorem 1.4Correct

Thickness for bounded trajectories of quasi-rays

Pages 3 and 11–18 · Theorem 1.4 and its proof · arXiv:2607.17021v1

The reduction to Theorem 3.1, the thinning argument for a quasi-ray, the quantitative equidistribution and tessellation estimates, and the strongly tree-like Cantor construction yield full local Hausdorff dimension in the expanding horospherical subgroup. Bounded thickening and the local product chart then give the stated thickness of $B(F)$ in $G/\Gamma$ . The quantifiers over the quasi-ray and nonempty open subsets are preserved.

Full paper, version 1 ↗

Theorem 3.1Correct

Uniform Cantor estimate for sufficiently interior quasi-rays

Pages 12–18 · Theorem 3.1 · arXiv:2607.17021v1

After the distance-from-walls reduction, the cited effective equidistribution estimate is uniform for the required increments. The chosen tessellation scale leaves a positive proportion of children at every stage, their diameters shrink exponentially, and the tree-like dimension bound tends to $\dim U$ as the scale parameter grows. These estimates establish the theorem in every required open subset.

Ronggang Shi, effective equidistribution input ↗

Theorem 1.5Correct

Weighted Diophantine application

Pages 5 and 18–23 · Theorem 1.5 and Section 4 · arXiv:2607.17021v1

Quasimultiplicativity supplies uniform positive increments, the weight functions are discretized into a quasi-ray in the super-expanding cone, and Proposition 4.4 gives the required Dani correspondence through Mahler's compactness criterion. Applying Theorem 1.4 and the local bi-Lipschitz parametrization of the unipotent subgroup gives the claimed thickness.

Related weighted transference input ↗

Power-weight exampleTypo

The exponent index should agree with the function index

Page 5 · Paragraph following Theorem 1.5 · arXiv:2607.17021v1

As printed, the standard example reads $\beta_j(T)=T^{b_i}$ . Replace $b_i$ by $b_j$ , giving $\beta_j(T)=T^{b_j}$ . The function is indexed by $j$ , the surrounding tuple is $(b_1,\ldots,b_n)$ , and no later step uses the mismatched index. The correction is unique, mechanical, and harmless through every downstream use.

02Proofs6 reported findingsCorrect

The main proofs and every material cited input checked are correct and complete. One genuine endpoint-domain mismatch has a unique verified local repair; its complete downstream scope was checked, so it changes no result and does not lower the overall proof status.

Theorem 1.4 via Theorem 3.1Correct and complete

Reduction, thinning, and local product argument

Pages 11–18 · Sections 2–3 · arXiv:2607.17021v1

The thinning argument, bounded-thickening implication, Cantor construction, and slicing step have the required quantifiers. Proposition 2.10 supplies separation from the cone walls and Theorem 3.1 gives full local dimension in $U$ . The set $B(F)$ is Borel: an exhaustion by compact sets writes it as a countable union of arbitrary intersections of closed preimages. The local product chart and slicing input therefore apply and transfer thickness to $G/\Gamma$ .

Kleinbock–Margulis bounded-orbit slicing precedent ↗

Shi effective-equidistribution inputCorrect and complete

The cited hypotheses match the manuscript's expanding-cone regime

Pages 8–10 · Theorem 2.5, Lemma 2.6, and Corollary 2.7 · arXiv:2607.17021v1

In Shi's theorem, $H$ is the normal product of the simple factors on which $U$ projects nontrivially; irreducibility and the absence of compact factors supply the required spectral gap; the acting element is nontrivial on every $H$ -factor; and its expanding horospherical subgroup is $U$ . For increments in the super-expanding cone, Lemma 2.6 identifies Shi's floor function with distance from the relevant cone wall. Corollary 2.7 therefore invokes the uniform compact-set error estimate in its stated regime.

Ronggang Shi, Expanding Cone and Applications to Homogeneous Dynamics ↗

Theorem 3.1 constructionCorrect and complete

Tessellation and strongly tree-like dimension input

Pages 12–18 · Lemma 3.3 and Propositions 3.4–3.6 · arXiv:2607.17021v1

The tessellation count, return estimate, positive-child condition, boundary-null intersections, shrinking diameters, and retained-density bound meet the stated tree theorem. Proposition 3.6 supplies at least the required positive proportion of children, while Lemma 3.3 gives exponential contraction. The rendered source uses compact closures of the tessellation pieces, so the compact-set hypotheses are met even though text extraction can omit the overlines.

Kleinbock–Margulis tessellation and slicing source ↗

Observation 4.2, Equation (4.5)Minor formal correction

The endpoint $t=0$ should be included

Page 19 · Observation 4.2, Equation (4.5); used on pages 20–22 · arXiv:2607.17021v1

Equation (4.5) is printed for $t>0$ , but the first increment in Observation 4.3 uses $t=(k-1)t_0=0$ when $k=1$ . Replace $t>0$ by $t\geq0$ , or state the endpoint extension separately. The proof of (4.5) works directly at $t=0$ : the relevant endpoint values are $\alpha_i(1)=\beta_j(1)=1$ , and the quasimultiplicative inequalities allow $RT=1$ for $\alpha_i$ and $T=1$ for $\beta_j$ . Repair classification: Verified repair. This fixes exactly the initial increment; every later increment is already in the printed domain.

Proposition 4.4Correct and complete

Generalized Dani correspondence

Pages 20–22 · Proposition 4.4 · arXiv:2607.17021v1

Normalization (4.3) gives determinant one, Mahler's compactness criterion applies, and vectors with $q=0$ have norm at least $1$ . Both implications correctly rescale the systems of inequalities using the lower and upper quasimultiplicative estimates. Omitting a bounded initial $t$ -interval is harmless by compactness and continuity.

Proof of Theorem 1.5Correct and complete

Application to weighted approximation

Page 23 · Proof of Theorem 1.5 · arXiv:2607.17021v1

Observation 4.3 produces exactly the quasi-ray required by Theorem 1.4, Proposition 4.4 identifies its bounded trajectories with the weighted badly approximable set, and the matrix-to-unipotent map is locally bi-Lipschitz. The hypotheses and parameter ranges of the three inputs match the final application.

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:2607.17021v1
Authors listed: Dmitry Kleinbock, Vasiliy Neckrasov
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 v1
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.

Bounded trajectories of quasi-rays on homogeneous spaces and Diophantine approximation with weight functions

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

Thickness for bounded trajectories of quasi-rays

Uniform Cantor estimate for sufficiently interior quasi-rays

Weighted Diophantine application

The exponent index should agree with the function index

Reduction, thinning, and local product argument

The cited hypotheses match the manuscript's expanding-cone regime

Tessellation and strongly tree-like dimension input

The endpoint t=0t=0t=0 should be included

Generalized Dani correspondence

Application to weighted approximation

Challenge an audit finding

Authenticate with ORCID, then require an exact arXiv match

The endpoint $t=0$ should be included