arXiv:2608.15845v1

Weighted singular vectors in common-base self-similar sets

Taehyeong Kim, Jaemin Park

math.NT11J1311J8328A80

Paper metadata

Original submission: August 16, 2026
Claimed MSC: 11J13, 11J83, 28A80
arXiv categories: math.NT
Paper license: arXiv non-exclusive distribution license ↗
Comments: 15 pages, 2 figures, 1 table

Abstract

We prove a lower bound for the Hausdorff dimension of weighted totally irrational singular vectors in affine-spanning common-base integral self-similar sets satisfying the open set condition. For the middle-third Cantor square, the bound improves the previously known explicit lower bound.

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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 19, 2026

01Statements3 reported findingsCorrect

The lower bound for weighted totally irrational singular vectors in affine-spanning common-base integral self-similar sets, and its middle-third Cantor-square specialization, are correct. A mechanical environment-name error in internal cross-references is typographical and does not affect either result.

Theorem 1.2Correct

Weighted lower bound on affine-spanning common-base self-similar sets

Pages 3 and 11 · Theorem 1.2 and its proof · arXiv:2608.15845v1

For $\alpha>w_1/(1-w_1)$ , the prescribed constant-symbol blocks give a set $E_\alpha\subset\operatorname{Sing}(\mathbf w)\cap K$ . The open set condition and the free-digit density give $\dim_HE_\alpha\ge h_{\mathbf w}(\alpha)\dim_HK,$ where $h_{\mathbf w}(\alpha)=\frac{\alpha(1-w_1)-w_1}{(\alpha-w_1)(1+\alpha)}.$ The affine-span hypothesis makes every rational affine hyperplane null for the digit-freezing measure, so deleting the non-totally-irrational points preserves this lower bound. The maximum over the admissible interval occurs at $\alpha_*=2w_1/(1-w_1)$ and equals $((1-w_1)/(1+w_1))^2$ , exactly as stated.

Theorem 1.1Correct

Weighted and unweighted lower bounds for the middle-third Cantor square

Pages 2 and 11 · Theorem 1.1 and its proof · arXiv:2608.15845v1

The base- $3$ digit system with $\mathcal D=\{0,2\}^2$ has attractor $C^2$ , satisfies the open set condition with $O=(0,1)^2$ , affinely spans $\mathbb R^2$ , and has $\dim_H(C^2)=2\log2/\log3$ . Substitution in Theorem 1.2 proves the weighted assertion. Setting $w_1=w_2=1/2$ gives the factor $1/9$ and hence $\dim_H\bigl(\operatorname{Sing}^*(2)\cap C^2\bigr)\ge\frac{2\log2}{9\log3},$ with no additional hypothesis or endpoint case.

Internal cross-referencesTypo

Shared-counter references print the wrong environment name

Pages 3, 5–7, and 9–11 · internal references to Definitions 2.1–2.2 and 2.5, Lemmas 2.4, 2.6, 2.8, and 3.1, and Corollaries 4.2–4.3 · arXiv:2608.15845v1

Several internal references print “Theorem” solely because the definitions, lemmas, propositions, and corollaries share the theorem counter. For example, Theorem 1.2 refers to “Theorem 2.1” and “Theorem 2.2,” which are Definitions 2.1 and 2.2, and its proof says “By Theorems 4.2 and 4.3,” which are Corollaries 4.2 and 4.3. Replace each printed environment name by the actual labeled environment name. The referenced numbers and mathematical targets are unambiguous, so the correction is unique, mechanical, and harmless.

02Proofs4 reported findingsCorrect

The cylinder count, free-digit Frostman estimate, hyperplane-null argument, constant-block approximation, parameter recursion, interval covering, density calculation, and final optimization are correct and complete. The only detected defect is the harmless internal cross-reference typo reported under Statements.

Lemmas 2.4 and 2.6Correct and complete

OSC cylinder counting and the free-digit dimension estimate

Pages 5–7 · Lemmas 2.4 and 2.6 · arXiv:2608.15845v1

Iterating the open set condition gives pairwise disjoint level- $N$ images of one fixed interior ball. Every cylinder meeting $B(x,b^{-N})$ places its interior ball inside a uniformly enlarged ball, so volume comparison gives a level-independent cylinder-count bound. A compatible length- $N$ symbolic cylinder has digit-freezing mass $(\#\mathcal D)^{-U_P(N)}$ . Combining these facts with $\#\mathcal D=b^{\dim_HK}$ yields, for every $\tau<\liminf U_P(N)/N$ , a uniform estimate $\mu_P(B(x,R))\ll R^{\tau\dim_HK}.$ The mass-distribution principle therefore gives the exact lower bound required for every later use.

Hutchinson, Fractals and Self-Similarity, Theorem 5.3 ↗

Lemma 2.8Correct and complete

Digit-freezing measures give zero mass to affine hyperplanes

Pages 7–9 · Lemma 2.8 · arXiv:2608.15845v1

Full affine span ensures that $u\cdot\mathcal D$ has at least two values for every $u\ne0$ . From infinitely many free positions one can select a subsequence with gaps at least $L$ , where the first differing projected digit dominates the entire projected tail. Hence a section of any affine hyperplane fixes at most one projected digit sequence. At each selected position its preimage has probability at most $m/\#\mathcal D<1$ , so every section has product measure zero and Fubini gives $\mu_P(H)=0$ . The countable union of rational affine hyperplanes is exactly the set of points for which $1,x_1,\ldots,x_n$ are rationally dependent, which proves the required almost-everywhere total irrationality.

Lemma 3.1 and Corollary 4.2Correct and complete

A constant-symbol block supplies the required weighted approximation

Pages 9–11 · Lemma 3.1 and Corollary 4.2 · arXiv:2608.15845v1

Repeating the fixed digit after a prefix of length $j$ produces the rational vector $\mathbf p/q$ with $q=(b-1)b^j$ . Agreement through a constant block of length $L$ gives $|qx_i-p_i|\le\Delta_{\mathcal D}b^{-L}$ . Since $w_n\le w_i\le w_1$ and $\Delta_{\mathcal D}\ge1$ , this implies $\max_i|qx_i-p_i|^{1/w_i}\le C_0b^{-L/w_1}.$ Whenever $q<T<(\varepsilon/C_0)b^{L/w_1}$ , the same pair satisfies the strict weighted-singularity inequalities. Proposition 4.1 covers every sufficiently large $T$ by one such interval, so every point of $E_\alpha$ is $\mathbf w$ -singular.

Proposition 4.1 and proof of Theorem 1.2Correct and complete

The block recursion, free-position density, and optimization close the proof

Pages 10–14 · Proposition 4.1, Section 5, and proof of Theorem 1.2 · arXiv:2608.15845v1

With $L_k=\lceil\alpha j_k\rceil$ and $j_{k+1}=\lfloor L_k/w_1\rfloor-\lceil\sqrt{j_k}\rceil$ , the hypothesis $\alpha>w_1/(1-w_1)$ makes the blocks disjoint and gives geometric growth. The overlap margin $L_k/w_1-j_{k+1}\asymp\sqrt{j_k}$ tends to infinity, so the approximation intervals cover all large $T$ for every fixed $\varepsilon>0$ . The minima of $U_\alpha(N)/N$ occur at the block endpoints. The limits $L_k/j_k\to\alpha$ and $j_{k+1}/j_k\to\alpha/w_1$ , followed by Stolz–Cesàro, yield precisely $h_{\mathbf w}(\alpha)$ . Differentiation gives the unique admissible maximizer $\alpha_*=2w_1/(1-w_1)$ ; restricting the Frostman measure to the full-measure totally irrational subset preserves each exponent below the endpoint and completes the Hausdorff-dimension argument.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper: arXiv:2608.15845v1
Authors listed: Taehyeong Kim, Jaemin Park
Audit date: August 19, 2026

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Weighted singular vectors in common-base self-similar sets

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

Weighted lower bound on affine-spanning common-base self-similar sets

Weighted and unweighted lower bounds for the middle-third Cantor square

Shared-counter references print the wrong environment name

OSC cylinder counting and the free-digit dimension estimate

Digit-freezing measures give zero mass to affine hyperplanes

A constant-symbol block supplies the required weighted approximation

The block recursion, free-position density, and optimization close the proof

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