MathAudit

Paper metadata

Original submission
December 22, 2009
Claimed MSC
11JXX
arXiv categories
math.NT
Comments
Paper in Russian, summary in English

Abstract

This paper is a survey of old and recent results related to Khintichine's singular matrices and their applications in the theory of Diophantine approximations. The paper is written in Russian. English version should appear in "Russian Mathematical Surveys" in the beginning of 2010.

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 20, 2026
01Statements4 reported findingsContains unsupported statements

This paper explicitly presents itself as a survey and attributes most reported results to named sources. Among the claims proved or derived in the paper, Theorem 12 and the exponent refinements in Section 5 are supported after mechanical notation corrections. Theorem 68 is not verified: only a sketch for dimension three is supplied, while the statement covers every dimension at least three.

Surveyed resultsCorrectly treated as cited results

Cited theorems are distinguished from present-paper claims

Pages 1–62 and 64–78 · attributed historical and recent results · arXiv:0912.4503v1

The abstract and introduction identify the article as a survey. Classical and modern results are normally accompanied by an author's name, a bibliography reference, or explicit wording that the result was announced elsewhere. Their omission of reproduced proofs is therefore not a proof defect of this paper. The audit's proof findings below concern the results for which this paper itself supplies a proof or derivation.

Theorem 12Correct

Almost-every extension preserves the eventual best-approximation sequence

Pages 17–19 · Theorem 12, condition (37), and proof · arXiv:0912.4503v1

For each competing integer vector with a nonzero added-coordinate block, the exceptional parameters form a product of nn slabs. Summing their measures over the original coordinates, added coordinates, and bounded integer shifts gives ζνnMν+1max⁡(m+n,m∗)(log⁡Mν+1)δ(m∗,m+n),\zeta_\nu^n M_{\nu+1}^{\max(m+n,m^*)}(\log M_{\nu+1})^{\delta(m^*,m+n)}, exactly the summand in condition (37). Borel–Cantelli then excludes all but finitely many competitors for almost every extension. Competitors whose added block vanishes are handled by the original best-approximation property. The notation corrections recorded below restore the intended exceptional set without changing the estimate.

Section 5 exponent refinementsCorrect

The determinant and rational-plane arguments support the new bounds

Pages 28–36 · Theorems 21, 23, and 25 and their proofs · arXiv:0912.4503v1

For the cases (m,n)=(1,3),(2,2),(3,1)(m,n)=(1,3),(2,2),(3,1), the proofs select maximal blocks of best approximations in a rational two-plane. The nonzero four-vector determinant gives the global alternative, while the covolume comparison inside the plane gives the complementary alternative. Substituting a power majorant and optimizing the auxiliary growth exponent yields the displayed functions g1g_1, g3g_3, and g2g_2. The determinant-row and function-name slips listed below are uniquely recoverable from their surrounding formulas.

Theorem 68Not able to verify

The arbitrary-gap singular-system construction is not fully proved

Pages 62–64 · Theorem 68, Lemma 8, and proof sketch · arXiv:0912.4503v1

The theorem asserts the construction for every m≥3m\geq3, every prescribed rapidly decreasing ψ\psi, and every prescribed rapidly increasing integer sequence τ(ν)\tau(\nu). The text explicitly explains only the case m=3m=3. Even there, Lemma 8 is supported only by a scheme that selects points inside a rational two-plane and then states that sufficiently small sector parameters make them exactly all ambient best approximations. No quantitative exclusion of integer competitors from the other lattice layers is supplied. The general case m>3m>3 is not addressed. These are central, nontrivial steps, and no complete repair was verified.

02Proofs7 reported findingsContains incorrect or incomplete proofs

The paper's Borel–Cantelli and exponent-inequality proofs close after uniquely determined notation repairs. The construction behind Theorem 68 remains incomplete both in its dimension-three extension lemma and in the passage to arbitrary dimension.

Proof of Theorem 12Correct and complete

The exceptional-set sum matches the convergence hypothesis

Pages 18–19 · Equations (39)–(41) · arXiv:0912.4503v1

After separating the zero added-coordinate block, a fixed competitor contributes O(ζνn/∥xm+1:m∗∥n)O(\zeta_\nu^n/\lVert x_{m+1:m^*}\rVert^n) in parameter measure. There are O(Mν+1m+n)O(M_{\nu+1}^{m+n}) choices of the original coordinates and integer shifts. Summing the added block by its maximum norm produces a power unless m∗=m+nm^*=m+n, when it produces the single logarithm encoded by δ(m∗,m+n)\delta(m^*,m+n). Thus the tail measure tends to zero under (37), which is precisely what the Borel–Cantelli step requires.

Theorem 12 notationTypo

Several symbols in the exceptional-set setup are mechanically reversed or omitted

Pages 17–19 · Theorem 12 and Equations (39)–(40) · arXiv:0912.4503v1

Equation (39) must read ∑iθjixi+yj\sum_i\theta_j^i x_i+y_j, and its minimization must exclude both adjacent lifted best approximations, not only the first. The point vector must contain generic yjy_j, not y1,νy_{1,\nu}; the theorem's final coordinate is yn,νy_{n,\nu}, not yn.νy_{n.\nu}. Both occurrences of JνJ_\nu require minus signs on every original term. Finally, Ων(x,y)\Omega_\nu(x,y) must use membership in JνJ_\nu for every row: it is the bad set where all errors are small, and the following small-measure bound is exactly for those slabs. The printed non-membership sign describes the complementary good set and cannot have the estimated measure. The stated goal and the immediately following estimate uniquely determine all these corrections.

Section 5 proofsCorrect and complete

The determinant/covolume case splits cover the required alternatives

Pages 28–36 · proofs of Theorems 21, 23, and 25 · arXiv:0912.4503v1

Each proof divides the growth of consecutive best-approximation norms into complementary ranges. A nonzero integral determinant supplies the first bound; constancy of the rank-two lattice covolume across the intervening block supplies the other. The monotone majorants requested in the text can be chosen without changing the limiting exponent. The optimization equations defining g1g_1, g3g_3, and g2g_2 then give the displayed lower bounds.

Section 5 notationTypo

Three local formulas contain uniquely identifiable symbol slips

Pages 28 and 32 and 34 · setup and proofs of Theorems 21, 23, and 25 · arXiv:0912.4503v1

The polynomial displayed for the endpoint α0\alpha_0 has a repeated xx term inconsistent with the immediately stated root interval and comparison and must be read with its intended constant term. In the determinant for the (2,2)(2,2) case, the third row's last entry must be y2,ky_{2,k} rather than x2,kx_{2,k}; the same proof's maximum runs over the two forms, not three. In the (3,1)(3,1) setup, h(α)=α−g2(α)−1h(\alpha)=\alpha-g_2(\alpha)-1, not the expression with undefined g(α)g(\alpha). Each correction is fixed by the theorem's dimension or by the function defined on the same page and has no effect on the status.

Lemma 8Incomplete as written · no verified repair supplied

Ambient best-approximation exclusion is only asserted

Pages 63–64 · Lemma 8 and its proof scheme · arXiv:0912.4503v1

The sketch chooses zc+1z_{c+1} in a neighboring lattice layer and zc+2,…,zdz_{c+2},\ldots,z_d as best approximations relative to a rational two-plane. It then says that sufficiently small η∗,δ∗\eta_*,\delta_* make these exactly the first dd best approximations for every parameter in the new sector. Establishing that assertion requires uniform separation from every competing point in all other lattice layers and preservation of the prescribed ordering throughout the sector. Neither a quantitative separation bound nor a finite-competitor reduction is given. The cited similar construction is not identified as a theorem that directly supplies this lemma.

Proof of Theorem 68Incomplete as written · no verified repair supplied

Only the case of three variables is discussed

Pages 62–64 · sentence preceding (107) and the induction after Lemma 8 · arXiv:0912.4503v1

The proof explicitly begins by restricting its explanation to m=3m=3, and its sectors, rational hyperplanes, and enumeration argument all live in R3\mathbb{R}^3 or R4\mathbb{R}^4. No reduction from general m≥3m\geq3 to this case and no higher-dimensional analogue of Lemma 8 is supplied. Since the theorem quantifies over every such mm, the omitted passage is a substantive proof obligation rather than a technical detail.

Theorem 69Correctly treated as cited, not as proved here

The following independence theorem is explicitly an external announcement

Page 64 · introduction and statement of Theorem 69 · arXiv:0912.4503v1

The paragraph immediately preceding Theorem 69 says that it is an assertion announced by Moshchevitin and German in reference [113]. The survey does not present it as a theorem proved in this paper, so the absence of a proof here is not counted as a defect in the paper's own proofs.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper
arXiv:0912.4503v1
Authors listed
Nikolay G. Moshchevitin
Audit date
August 20, 2026
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