arXiv:2408.06200v3

Dirichlet improvability in $L_p$ -norms

Nikolay Moshchevitin, Nikita Shulga

math.NTmath.DS11J8311H0611J70

Paper metadata

Original submission: August 12, 2024
Latest revision: June 22, 2026 · version 3
Claimed MSC: 11J83, 11H06, 11J70
arXiv categories: math.NT, math.DS
Paper license: CC BY 4.0 ↗
Comments: 34 pages plus 13 pages long Appendix, any comments are appreciated

Abstract

For a norm $F$ on $\mathbb{R}^2$ , we consider the set of $F$ -Dirichlet improvable numbers $\mathbf{DI}_F$ . In the most important case of $F$ being an $L_p$ -norm with $p=\infty$ , which is a supremum norm, it is well-known that $\mathbf{DI}_F = \mathbf{BA}\cup \mathbb{Q}$ , where $\mathbf{BA}$ is a set of badly approximable numbers. It is also known that $\mathbf{BA}$ and each $\mathbf{DI}_F$ are of measure zero and of full Hausdorff dimension. Using classification of critical lattices for unit balls in $L_p$ , we provide a complete and effective characterization of $\mathbf{DI}_p:=\mathbf{DI}_{F^{[p]}}$ in terms of the occurrence of patterns in regular continued fraction expansions, where $F^{[p]}$ is an $L_p$ -norm with $p\in[1,\infty)$ . This yields several corollaries. In particular, we resolve two open questions by Kleinbock and Rao by showing that the set $\mathbf{DI}_{p}\setminus \mathbf{BA}$ is of full Hausdorff dimension, as well as proving some results about the size of the difference $\mathbf{DI}_{p_1}\setminus \mathbf{DI}_{p_2}$ . To be precise, we show that the set difference of Dirichlet improvable numbers in Euclidean norm ( $p=2$ ) minus Dirichlet improvable numbers in taxicab norm ( $p=1$ ) and vice versa, that is $\mathbf{DI}_{2}\setminus \mathbf{DI}_{1}$ and $\mathbf{DI}_{1}\setminus \mathbf{DI}_{2}$ , are of full Hausdorff dimension. We also find all values of $p$ , for which the set $\mathbf{DI}_p^c\cap\mathbf{BA}$ has full Hausdorff dimension. Finally, our characterization result implies that the number $e$ satisfies $e\in \mathbf{DI}_p$ if and only if $p\in(1,2)\cup(p_0,\infty)$ for some special constant $p_0\approx2.57$ .

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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

01Statements2 reported findingsContains unsupported statements

The continued-fraction classification and the Hausdorff-dimension consequences are internally coherent conditional on the asserted classification of critical lattices for $L_p$ balls. For $1<p<6$ , $p\neq2$ , that material input rests on a computer-assisted verification whose decisive formulas and certificate are not contained in a fixed reviewed artifact, so the full all- $p$ statements are not able to be verified from the available evidence. No counterexample to any stated theorem was found.

Theorem 3.1Not able to verify

The all- $p$ classification depends on an unverified computer-assisted input

Pages 8–10 and 14–20 · Theorem 3.1 and Sections 5–6 · arXiv:2408.06200v3

The theorem classifies $\mathbf{DI}_p^c$ for every $p\in[1,\infty)$ by first identifying every critical lattice of the $L_p$ unit ball. The manuscript itself records that the published proof of this critical-lattice classification contains false derivative-sign assertions, and Appendix A replaces it, for $1<p<6$ and $p\neq2$ , by interval computations. The unresolved obligation is the uniform inequality $\Delta(p,\sigma)>\min\{\Delta(p,1),\Delta(p,\sigma_p)\}$ throughout that full two-parameter region. Appendix A supplies tables and three pieces of pseudocode, but omits several analytic derivative expressions, refers those expressions to mutable external code, gives no fixed commit or independently checkable output certificate, and describes the fuller verification as work in preparation. I was therefore unable to verify the critical-lattice input in these regimes. The classical $p=1,2$ descriptions and the cited analytic range $p\geq6$ are not affected by this finding, and no evidence that the classification is false was found.

Computer-assisted verification repository cited as reference 43 ↗

Theorem 1.1Not able to verify

The full parameter-range dimension conclusion inherits the unresolved input

Page 4 · Theorem 1.1; pages 24–25 · its proof · arXiv:2408.06200v3

Theorem 1.1 claims $\dim_H(\mathbf{DI}_p\setminus\mathbf{BA})=1$ for every finite $p\geq1$ . Its sparse-insertion argument and dimension-preservation lemma support the conclusion once Theorem 3.1 is available. However, the proof explicitly invokes Theorem 3.1 to decide which inserted continued-fraction patterns are excluded or forced. Thus the all- $p$ portion of Theorem 1.1 is not independently verified in the same regimes as the preceding finding. Theorem 1.2 uses only the classical $p=1,2$ critical loci and is supported by the internal argument, so no adverse conclusion is drawn about it.

Full paper, version 3 ↗

02Proofs3 reported findingsContains unverified proofs

The compactness-to-pattern reduction and the sparse-insertion dimension arguments are verifiable, conditional on the critical-lattice classification. The interval-arithmetic replacement for the flawed historical proof is not fully auditable from the paper and its unpinned external code, leaving a precise foundational proof obligation unresolved rather than disproved.

Appendix ANot able to verify

The interval verification is not self-contained or fixed to a reproducible certificate

Pages 33–45 · Appendix A and Appendix B · arXiv:2408.06200v3

The appendix correctly isolates the needed inequality and partitions the domain into explicit numerical boxes, but Remark A.3 withholds the full formulas for several differentiated functions, Tables 1–4 report claimed checks rather than enclosing intervals or machine-verifiable certificates, and the cited repository is not pinned to a commit. This prevents checking that the implemented expressions equal the mathematical derivatives, that every branch of the implicitly defined $\tau(p,\sigma)$ is enclosed, and that every box in the open region is covered without overlap gaps. These are concrete nontrivial obligations because the paper documents counterexamples to corresponding sign assertions in references 13–14. Repair classification: provide a fixed source archive and environment, complete formulas or generated-expression hashes, certified run logs covering every listed box, and a coverage proof tying the boxes to Equation (57).

Full paper, version 3 ↗

Proposition 4.1 and proof of Theorem 3.1Correct and complete

The compactness and continued-fraction reduction is correct conditional on the critical locus

Pages 12–20 · Proposition 4.1, Lemma 6.1, and proof of Theorem 3.1 · arXiv:2408.06200v3

Mahler compactness converts equality in the Dirichlet bound into accumulation on the normalized critical locus. In each listed critical-lattice type, the two distinguished lattice vectors become consecutive best-approximation vectors by Lemma 6.1; the ratios in Equation (7) then force exactly the finite, rational-tail, or infinite palindromic patterns stated in Theorem 3.1. Conversely, a sequence of such patterns fixes both basis-vector ratios and, after determinant normalization, produces the required critical-lattice accumulation point. This part of the proof introduces no additional defect beyond the availability of the complete critical locus itself.

Lemma 9.1 and Sections 8–10Correct and complete

The dimension-preserving insertion arguments are correct

Pages 21–32 · Lemma 8.1, Lemma 9.1, and proofs of Theorems 1.1–1.2 · arXiv:2408.06200v3

The inserted symbols have density $\omega(n)=o(n^{1-\varepsilon_1})$ , while condition (2) bounds their size polynomially. Continuant estimates therefore make deletion locally $1/(1+\varepsilon)$ -Hölder and insertion locally Lipschitz on every bounded-partial-quotient stratum. Taking the countable union over those strata preserves Hausdorff dimension. The chosen sparse words then either force or avoid the required critical patterns, and their unbounded entries exclude bad approximability. The $p=1,2$ comparison in Theorem 1.2 is consequently verified independently of the computer-assisted nonclassical critical loci.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper: arXiv:2408.06200v3
Authors listed: Nikolay Moshchevitin, Nikita Shulga
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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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. 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P(delta)

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t>0

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Dirichlet improvability in LpL_pLp​-norms

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

The all-ppp classification depends on an unverified computer-assisted input

The full parameter-range dimension conclusion inherits the unresolved input

The interval verification is not self-contained or fixed to a reproducible certificate

The compactness and continued-fraction reduction is correct conditional on the critical locus

The dimension-preserving insertion arguments are correct

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Dirichlet improvability in $L_p$ -norms

The all- $p$ classification depends on an unverified computer-assisted input