arXiv:2607.24161v1

Slowly Divergent Trajectories for Weighted Singular Vectors in R^2

Bohan Yang

math.NTmath.DS11J1311K5537A17

Paper metadata

Original submission: July 27, 2026
Claimed MSC: 11J13, 11K55, 37A17
arXiv categories: math.NT, math.DS
Paper license: arXiv non-exclusive distribution license ↗
Comments: 28 pages

Abstract

Let $w=(w_1,w_2)$ satisfy $w_1>w_2>0$ and $w_1+w_2=1$ . For every function $f:[0,\infty)\to(0,\infty)$ with $f(t)\to0$ , we prove that the set of $w$ -singular vectors whose weighted shortest-vector function satisfies $W_x(t)\ge\log f(t)$ for all sufficiently large $t$ has Hausdorff dimension $s_w=2-(1+w_1)^{-1}$ , equal to the full Hausdorff dimension of $\operatorname{Sing}_w(2)$ . For every $0<ν<1$ and $μ>0$ , a separate power schedule gives Hausdorff dimension $s_w$ for the weighted uniform approximation set with rate $Q^{-1}\exp\bigl(-μ(\log Q)^ν\bigr)$ , whereas the lower-envelope theorem gives the same dimension for its complement in $\operatorname{Sing}_w(2)$ . We give a direct proof of the lower-envelope theorem by adapting the self-affine construction of Liao--Shi--Solan--Tamam to a variable sequence of return times and using an empty-denominator-window argument to control intermediate cusp excursions without further pruning the tree.

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

01Statements3 reported findingsCorrect

The arbitrary lower-envelope theorem for weighted singular vectors, both dimension assertions for the prescribed subpower uniform-approximation rate, and the full-lattice consequence have the stated scopes and follow from the proved construction and the cited ambient-space dimension bounds.

Theorem 1.1Correct

Arbitrarily slow weighted divergence

Page 2 · Theorem 1.1; proof on pages 23–24 · arXiv:2607.24161v1

For an arbitrary positive $f$ with $f(t)\to0$ , Lemma 2.2 replaces it by a divergent nondecreasing $1$ -Lipschitz target $\psi$ satisfying $e^{-\psi(t)}\geq f(t)$ . The variable-step tree uses $H_n=\psi(t_n)$ , $d_{n+1}=H_n^{1/2}$ , and $\varepsilon_n=e^{-H_n^{1/4}}$ . Proposition 4.2 verifies all counting, nesting, separation, shrinking, and singularity hypotheses and makes the denominator interval in Lemma 4.1 empty. Hence every nonexceptional branch limit satisfies $W_x(t)\geq-\psi(t)$ eventually. Proposition 3.13 gives dimension $s_w=2-(1+w_1)^{-1}$ , while the known dimension of $\operatorname{Sing}_w(2)$ gives the matching upper bound.

Liao–Shi–Solan–Tamam, weighted singular-vector dimension ↗

Theorem 1.2Correct

Subpower uniform-approximation rate and its complement

Page 3 · Theorem 1.2; proof on pages 24–27 · arXiv:2607.24161v1

For $0<\nu<1$ and $\mu>0$ , the power schedule $d_{n+1}=t_n^{(1+\nu)/2}$ and $\varepsilon_n=e^{-\chi t_n^\nu}$ , with $\chi>w_1\mu$ , retains the full branching dimension. The vertical return at level $n$ supplies an approximant with $q_n\leq e^{t_n}$ and weighted error at most $\varepsilon_n^{1/w_1}e^{-t_{n+1}}$ . Uniformly for $e^{t_n}\leq Q<e^{t_{n+1}}$ , this is at most $Q^{-1}e^{-\mu(\log Q)^\nu}$ for all large $n$ . Conversely, Corollary 6.1 applies Theorem 1.1 to a target $h(t(Q))=2\sqrt{Q\Phi(Q)}$ , where $t(Q)=\frac12\log(Q/\Phi(Q))$ , and obtains a full-dimensional subset with $\Delta_x(Q)\geq2\Phi(Q)$ eventually. Both sets lie in $\operatorname{Sing}_w(2)$ , so the upper dimensions are also $s_w$ .

Kleinbock–Moshchevitin–Warren–Weiss, weighted uniform-approximation framework ↗

Corollary 1.3Correct

Full-lattice lower-envelope dimension

Page 3 · Corollary 1.3 and the paragraph following it · arXiv:2607.24161v1

The construction is uniform for the compact family $v_s\mathbb Z^3$ and therefore gives $s_w$ dimensions on each two-dimensional $u_x$ -slice. Slicing in the remaining expanding root direction adds one dimension. The five central or contracting local-product directions remain uniformly bounded after conjugation by $a_t$ ; the elementary comparison of Euclidean and weighted minima explains the paper's use of the stronger seed envelope $c f(t)^{w_2}$ and transfers the desired envelope $f$ to those products. This yields the lower bound $8-(1+w_1)^{-1}$ . Solan's general-flow estimate gives the same dimension as an upper bound for the ambient divergent set, which contains the set in the corollary.

Solan, general-flow parametric geometry of numbers ↗

02Proofs4 reported findingsCorrect

The proof chains are correct and complete. The balance-time reduction has the required quantifiers, the variable-step modification preserves the cited LSST count and self-affine dimension estimate, the global schedule excludes every possible intermediate bad excursion, and the two uniform-rate arguments use the weighted error functional with the correct inequality directions.

Proposition 2.1 and Lemma 2.2Correct and complete

Balance functions and target regularization

Pages 3–5 · Section 2 · arXiv:2607.24161v1

For a primitive vector with $q>0$ and $R_x(v)>0$ , its contribution is exactly $m_x(v)+|t-\tau_x(v)|$ . Monotonicity and the $1$ -Lipschitz property give $|t-\tau|+\psi(t)\geq\psi(\tau)$ , while vectors with bounded balance time have $R_x(v)$ bounded away from zero by lattice discreteness and therefore cannot affect the eventual lower envelope. The infimal $1$ -Lipschitz regularization is nonnegative, nondecreasing, divergent, and satisfies $e^{-\psi}\geq f$ , with every endpoint and eventual quantifier preserved.

Full paper, version 1 ↗

Theorem 3.9 and Proposition 3.13Correct and complete

Variable-step counting and dimension

Pages 7–19 · Sections 3.2–3.5 · arXiv:2607.24161v1

Lemma 3.6 bijects candidate centers with primitive points in the required shell. Lemmas 3.7–3.8 put the three comparison boxes within the cited primitive lattice-point estimate. Lemma 3.5 injects each illegal class into the corresponding LSST exceptional set; after choosing $r$ , $\varepsilon_*$ , and $d_*$ in that order, their total is a fixed fraction of the $\varepsilon_n^2e^{2d_n}$ main term. The dual-systole condition supplies sibling separation. In the dimension calculation, $\log P_n=2t_n-o(t_n)$ , the long/short crossover obeys $t_{D_n}/t_n\to(1+w_1)/(1+w_2)$ , and the separation loss is $o(t_n)$ , giving every $s<s_w$ in the quoted self-affine criterion.

Liao–Shi–Solan–Tamam, source of the quoted counting and self-affine estimates ↗

Lemma 4.1 and Proposition 4.2Correct and complete

Denominator localization and empty windows

Pages 19–23 · Section 4 · arXiv:2607.24161v1

The parent dual-systole bound gives a uniform lower bound for the Euclidean first minimum at time $t_n$ . If a vector balancing in $[t_n,t_{n+1}]$ had $m_x(v)<\log\alpha_n$ , this forces $12^{-1/w_2}\varepsilon_n^{2/w_2}e^{t_n}\leq q<\alpha_ne^{t_{n+1}}.$ With $\alpha_n=e^{-H_n}$ , $\varepsilon_n=e^{-H_n^{1/4}}$ , and $t_{n+1}-t_n=H_n^{1/2}$ , inequality (4.5) makes the lower endpoint strictly exceed the upper endpoint. The $1$ -Lipschitz bound on $\psi$ simultaneously verifies all recursive hypotheses and the asymptotics required for divergence and dimension.

Full paper, version 1 ↗

Corollary 6.1 and Lemmas 6.2–6.3Correct and complete

Rate avoidance and attainment

Pages 24–27 · Section 6 · arXiv:2607.24161v1

At $t(Q)=\frac12\log(Q/\Phi(Q))$ , the vertical term for every $q\leq Q$ is at most half of $h(t(Q))=2\sqrt{Q\Phi(Q)}$ ; hence an $h$ -lower envelope forces $R_x(v)\geq e^{-t(Q)}h(t(Q))=2\Phi(Q)$ . For attainment, Lemma 6.2 correctly takes $1/w_i$ powers of the coordinate bounds and uses $w_1>w_2$ and $0<\varepsilon_n\leq1$ to obtain the common factor $\varepsilon_n^{1/w_1}$ . The choice $(1+\nu)/2$ lies strictly between $\nu$ and $1$ , so all tree errors are sublinear in the return step and $t_{n+1}/t_n\to1$ , exactly as required.

Full paper, version 1 ↗

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:2607.24161v1
Authors listed: Bohan Yang
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.

Slowly Divergent Trajectories for Weighted Singular Vectors in R^2

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

Arbitrarily slow weighted divergence

Subpower uniform-approximation rate and its complement

Full-lattice lower-envelope dimension

Balance functions and target regularization

Variable-step counting and dimension

Denominator localization and empty windows

Rate avoidance and attainment

Challenge an audit finding

Authenticate with ORCID, then require an exact arXiv match