Published paper

Khintchine's singular Diophantine systems and their applications

Nikolay G. Moshchevitin

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Publication metadata

Journal citation: Russian Mathematical Surveys 65(3), 433–511 (2010)
Publication date: 2010
DOI: 10.1070/RM2010v065n03ABEH004680 ↗
arXiv counterpart: arXiv:0912.4503v1

Abstract

The survey proves the best-approximation dimension dichotomy used to deduce the focal paper's four-dimensional corollary.

Role in dependence graphs

Proof-critical source

On Some Properties of Irrational Subspaces

This paper is included only for the following marked statement:

Theorem 7, Corollary 4 · best-approximation sectionProvides the two-or-four dimensional alternative for the eventual span of best approximations in the $2 imes2$ case.

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Russian Mathematical Surveys 65(3) (2010), 433-511 (English version of record)

Nikolay G. Moshchevitin, "Khintchine's singular Diophantine systems and their applications," Russian Mathematical Surveys 65(3) (2010), 433-511.

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Generated August 23, 2026

01Statements4 reported findingsContains unsupported statements

The article explicitly presents itself as a survey and properly attributes most reported results. Its proved extension theorem and exponent refinements are supported after local notation corrections. The arbitrary-dimensional construction in Theorem 68 is not verified because the published text gives only a sketch for dimension three and leaves its key ambient exclusion step unproved.

Surveyed resultsCorrectly treated as cited results

Cited theorems are distinguished from present-paper claims

Russian Mathematical Surveys 65 (2010), pp. 433-496 and 499-511, attributed historical and recent results

The abstract and introduction identify the work as a survey. Classical and modern results are normally accompanied by an author's name, a bibliographic reference, or explicit wording that the result was announced elsewhere. The absence of reproduced proofs for those attributed statements is therefore not treated as a defect in the paper's own arguments.

Theorem 12Correct

Almost every extension preserves the eventual best-approximation sequence

Russian Mathematical Surveys 65 (2010), pp. 450-452, Theorem 12 and equations (37)-(41)

For each competitor with a nonzero added-coordinate block, the exceptional parameters form a product of $n$ slabs. Summing their measures over the original coordinates, added coordinates, and bounded integer shifts gives the series in (37). Borel-Cantelli then excludes all but finitely many competitors for almost every extension. Competitors with zero added block are controlled by the original best-approximation sequence.

Section 5 exponent refinementsCorrect

The determinant and rational-plane arguments support the new bounds

Russian Mathematical Surveys 65 (2010), pp. 461-468, Theorems 20, 22, and 24 and their proofs

In the cases $(m,n)=(1,3),(2,2),(3,1)$ , the proofs select maximal blocks of best approximations in a rational two-plane. A nonzero four-vector determinant gives one growth alternative, while the rank-two lattice covolume gives the complementary alternative. Substitution of a power majorant and optimization of the auxiliary growth exponent yields the displayed functions $g_1$ , $g_3$ , and $g_2$ .

Theorem 68Not able to verify

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

Russian Mathematical Surveys 65 (2010), pp. 497-499, Theorem 68, Lemma 8, and the proof sketch

The theorem quantifies over every $m\geq3$ , every prescribed decreasing function $\psi$ , and every increasing sequence $\tau(\nu)$ . The text explicitly sketches only $m=3$ . Even there, Lemma 8 asserts without a quantitative argument that sufficiently small sector parameters make the selected relative best approximations exactly all ambient best approximations. The general case $m>3$ is not addressed, and no complete repair was verified.

02Proofs7 reported findingsContains incorrect or incomplete proofs

The Borel-Cantelli argument and the three exponent-refinement proofs close after uniquely determined notation repairs. The construction supporting Theorem 68 remains incomplete both in its dimension-three extension lemma and in its passage to arbitrary dimension.

Proof of Theorem 12Correct and complete

The exceptional-set sum matches the convergence hypothesis

Russian Mathematical Surveys 65 (2010), pp. 451-452, equations (39)-(41)

After separating the zero added-coordinate block, a fixed competitor contributes $O(\zeta_\nu^n/\lVert x_{m+1:m^*}\rVert^n)$ in parameter measure. Counting original coordinates and integer shifts, then summing the added block by maximum norm, produces the power $M_{\nu+1}^{\max(m+n,m^*)}$ and exactly one logarithm in the equality case. This is the summand in (37).

Theorem 12 notationTypo

The exceptional-set setup contains mechanically repairable symbol slips

Russian Mathematical Surveys 65 (2010), pp. 451-452, equations (39)-(41)

The competitor vector in the line following (39) must use generic $y_1$ rather than $y_{1,\nu}$ , and the exclusion must cover the adjacent lifted best approximations used immediately below. Most decisively, $\Omega_\nu(x,y)$ must require membership in every interval $J_\nu$ , not non-membership: it is the bad set where all errors are small, and only that set has the following $O(\zeta_\nu^n)$ measure estimate. The stated goal and the estimate uniquely determine these local corrections; the mathematical argument is unchanged.

Theorems 20, 22, and 24Correct and complete

The determinant and covolume case splits cover all required alternatives

Russian Mathematical Surveys 65 (2010), pp. 461-468, proofs of Theorems 20, 22, and 24

Each proof partitions the growth of consecutive best-approximation norms into complementary ranges. A nonzero integral determinant supplies the first estimate and the constant covolume of the rank-two lattice across the intervening block supplies the other. The monotone majorants can be chosen as stated, and the optimization equations for $g_1$ , $g_3$ , and $g_2$ yield the displayed lower bounds.

Section 5 notationTypo

Two published formulas retain uniquely identifiable dimension and function-name slips

Russian Mathematical Surveys 65 (2010), pp. 466-467, proof of Theorem 22 and cases 2-3 in the proof of Theorem 24

On p. 466 the maximum in the empty-parallelepiped display must run over the two forms, not three. On p. 467 the exponents in cases 2 and 3 use an undefined $g(\alpha(\Theta))$ ; the function defined on the preceding page and used in the closing polynomial identity is $g_2(\alpha(\Theta))$ . Each repair is forced by the dimension or the immediately adjacent definition and does not alter the proof.

Lemma 8Incomplete as written - no verified repair supplied

Ambient best-approximation exclusion is only asserted

Russian Mathematical Surveys 65 (2010), pp. 497-498, Lemma 8 and its proof sketch

The sketch chooses $z_{c+1}$ in a neighboring lattice layer and $z_{c+2},\ldots,z_d$ as best approximations relative to a rational two-plane, then says small $\eta_*$ and $\delta_*$ make these exactly the first $d$ ambient best approximations throughout the new sector. That requires uniform separation from every competing point in all other lattice layers and preservation of the prescribed ordering. Neither a quantitative separation bound nor a finite-competitor reduction is supplied.

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

Only the case of three variables is discussed

Russian Mathematical Surveys 65 (2010), pp. 497-499, proof following Theorem 68

The proof explicitly restricts its explanation to $m=3$ , and its sectors, rational hyperplanes, and enumeration argument all live in $\mathbb{R}^3$ or $\mathbb{R}^4$ . No reduction from general $m\geq3$ to this case and no higher-dimensional version of Lemma 8 is supplied. The omitted passage is a substantive proof obligation.

Theorem 69Correctly treated as cited, not as proved here

The following independence theorem is explicitly external

Russian Mathematical Surveys 65 (2010), p. 499, paragraph preceding Theorem 69

The text says that Theorem 69 was announced in reference [34]. The survey does not present a proof, so its absence is not counted as a defect in the paper's own arguments.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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