arXiv:2609.35009v1

Hausdorff Dimension of Weighted Singular Vectors

math.NTmath.DS11J1311J8328A78

Paper metadata

Original submission: September 28, 2026
Claimed MSC: 11J13, 11J83, 28A78
arXiv categories: math.NT, math.DS
Paper license: arXiv non-exclusive distribution license ↗
Comments: 18 pages

Abstract

Let $d\ge2$ and let $\mathbf w=(w_1,\ldots,w_d)$ satisfy $w_1\ge\cdots\ge w_d>0$ and $\sum_i w_i=1$ . Set $s_*=d-(1+w_1)^{-1}$ . We prove that there exist constants $C_{d,\mathbf w}>0$ and $\varepsilon_0=\varepsilon_0(d,\mathbf w)>0$ such that for all $0<\varepsilon<\varepsilon_0$ , $\dim_H\operatorname{DI}_{\mathbf w}(\varepsilon)\le s_*+C_{d,\mathbf w}\sqrt\varepsilon.$ Together with the lower bound of Kim--Park, this gives the exact formula $\dim_H\operatorname{Sing}(\mathbf w)=s_*.$

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Audited against arXiv v1

Not a correctness certificate. A green “no issues found” result may include yellow suspected typos or minor formal issues 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.

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Detailed mathematical audit

Generated October 24, 2026

01Statements3 reported findingsNo statement issues found

The accelerated best-approximation construction proves the stated $O_{d,\mathbf w}(\sqrt{\varepsilon})$ upper bound for weighted Dirichlet-improvable vectors. Together with the cited Kim–Park lower bound, it gives the exact Hausdorff dimension $d-(1+w_1)^{-1}$ for weighted singular vectors.

Theorem 1.2No issue found

The quantitative dimension upper bound follows from a contracting accelerated cover

PDF pp. 3–14 · arXiv:2609.35009v1

Late best approximations of a point in $\operatorname{DI}_{\mathbf w}(\varepsilon)$ lie in $Q_\xi$ with $\xi=c_{\mathbf w}\varepsilon$ . A gap in their successive minima supplies a proper rational subspace and an exit whose level grows by at least $R$ . The transverse lattice comparison and shell counts give a successor sum below one at exponent $d+\delta$ with $\delta=M\sqrt\xi$ . Covering each approximation rectangle at its shortest side then converts this exponent into $s_\ast+\delta/(1+w_1)$ , exactly the claimed error order.

Theorem 1.1No issue found

The exact weighted singular dimension is obtained

PDF pp. 2–3 and 14–16 · arXiv:2609.35009v1

Since $\operatorname{Sing}(\mathbf w)\subseteq\operatorname{DI}_{\mathbf w}(\varepsilon)$ for every $\varepsilon$ , Theorem 1.2 gives the upper bound $s_\ast$ after letting $\varepsilon\downarrow0$ . The lower bound $s_\ast$ is precisely the theorem stated in the cited Kim–Park paper (arXiv:2207.07944v2). The independent minimal-vector argument in Appendix A also proves the required upper bound directly.

Appendix ANo issue found

The direct singular-dimension proof gives the same sharp upper bound

PDF pp. 14–17 · arXiv:2609.35009v1

Minimality supplies a uniform separation in the rescaled Farey lattice. The one-step lattice count has only a logarithmic loss in $1/\varepsilon$ , and iterating the resulting cover gives the error $\log H_\varepsilon/((1+w_1)\log(1/\varepsilon))$ . This tends to zero and recovers $\dim_H\operatorname{Sing}(\mathbf w)\leq s_\ast$ .

02Proofs4 reported findingsNo proof issues found

The best-approximation estimates, successive-minimum gap construction, quotient-lattice comparison, shell counting, and Hausdorff-cover recursion were checked through their proof-critical inequalities; no substantive gap was found.

Lemmas 2.1–2.6No issue found in the reviewed proof

The acceleration is available in every dimension

PDF pp. 4–7 · arXiv:2609.35009v1

The Farey lattice has covolume $|u|^{-1}$ and its shortest vector controls the weighted best-approximation cell with the stated constant. Minkowski's second theorem forces a multiplicative gap among the successive minima when $\xi$ is small. The span below that gap lifts to a proper rational subspace $H_u$ , and Lemma 2.6 shows that the first later best approximation outside $H_u$ has level greater than $RL_u|u|$ . Rational independence guarantees that an exit eventually occurs.

Lemmas 3.1–3.3 and Proposition 3.4No issue found in the reviewed proof

Transverse minima and exit counts have the required powers

PDF pp. 7–12 · arXiv:2609.35009v1

Modulo $V_u$ , the rescaled lattices $D_{|u|/|v|}\Gamma_v$ and $\Gamma_u$ coincide. Reducing coefficients against a lattice basis of $V_u$ costs at most $m_uL_u/2$ , while the gap $\lambda_{m_u+1}>RL_u$ absorbs that cost and yields the factors $(|v|/|u|)^{w_d}$ . Counting projected lattice points in the intermediate and exit shells then gives exactly the three cases in Proposition 3.4; the injectivity assertions hold because each level interval has the length of the relevant primitive denominator.

Proposition 4.1 and Lemma 4.2No issue found in the reviewed proof

The successor sum contracts at the dimension exponent

PDF pp. 12–14 · arXiv:2609.35009v1

After multiplying the witness count by the exit count and the weight $(|u|/|w|)^{d+\delta}$ , all $k$ -series converge uniformly and the two possible $\ell$ -series cost at most $O(\delta^{-1})$ . The transverse-minimum product contributes $O_R(\sqrt\xi)$ and the final minimum contributes $O(R^{-1})$ . Choosing $R$ first and then $\delta=M\sqrt\xi$ makes the sum strictly contractive. The cube count for $\beta(u)$ has total $s$ -cost $|u|^{-d-(1+w_1)(s-s_\ast)}$ , matching the successor exponent without a missing factor.

Lemma A.1No issue found in the reviewed proof

The minimal-vector separation and recursive cover are coherent

PDF pp. 14–16 · arXiv:2609.35009v1

A lattice vector of sup norm below $1/2$ would, after reducing its level modulo $q$ and primitively rescaling, contradict minimality. Full-dimensional lattice spans contribute $O(\varepsilon k^d/N)$ points and proper spans contribute $O(k^{d-1})$ ; weighting by the level gives the harmonic loss $1+\log(1/\varepsilon)$ . The subsequent recursion and cube side $\varepsilon^{w_1}T_n^{-1-w_1}$ yield the displayed dimension exponent.

03Novelty0 reported findingsNo novelty concerns found

No non-novelty findings.

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Paper: arXiv:2609.35009v1
Authors listed: Bohan Yang, Tianru Zhu
Audit date: October 24, 2026

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Hausdorff Dimension of Weighted Singular Vectors

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

The quantitative dimension upper bound follows from a contracting accelerated cover

The exact weighted singular dimension is obtained

The direct singular-dimension proof gives the same sharp upper bound

The acceleration is available in every dimension

Transverse minima and exit counts have the required powers

The successor sum contracts at the dimension exponent

The minimal-vector separation and recursive cover are coherent

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Authenticate with ORCID, then require an exact arXiv match