MathAudit

Paper metadata

Original submission
July 16, 2021
Claimed MSC
11J13
arXiv categories
math.NT
Comments
14 pages
Journal reference
Unif. Distrib. Theory 17 (2022), no.1, 89 - 104

Abstract

In this paper we discuss some properties of completely irrational subspaces. We prove that there exist completely irrational subspaces that are badly approximable and, moreover, sets of such subspaces are winning in different senses. We get some bounds for Diophantine exponents of vectors that lie in badly approximable subspaces that are completely irrational; in particular, for any vector ξξ from two-dimensional badly approximable completely irrational subspace of Rd\mathbb{R}^d one has ω^(ξ)≤5−12\hatω(ξ) \leq \frac{\sqrt{5} - 1}{2}. Besides that, some statements about the dimension of subspaces generated by best approximations to completely irrational subspace easily follow from properties that we discuss.

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 19, 2026
01Statements3 reported findingsCorrect

The central winning and hyperplane-absolute-winning results for completely irrational subspaces, the upper bound for the uniform exponent on a badly approximable completely irrational subspace, and the lower bound for the eventual span of best approximations are correct. The printed proofs contain a false Taylor formula and several uniquely repairable notation errors, but the repairs below verify the stated conclusions.

Theorems 2.1 and 2.3Correct

Completely irrational matrices form winning and hyperplane-absolute-winning sets

Pages 6–9 · algebraic-manifold escaping argument and its HAW analogue · arXiv:2107.08084v1

For each rational complementary-dimensional subspace, nontrivial intersection with the graph of a matrix is the zero set of a nonzero determinant polynomial in the matrix entries. After replacing the printed Taylor identity by the correct finite multivariable Taylor expansion, the escaping argument forces a later game ball a positive distance from that zero set. Scheduling the countably many rational subspaces successively preserves all earlier avoidances. The same induction works in the hyperplane absolute game, where Alice removes a neighborhood of the affine hyperplane supplied by the nonzero gradient. Thus the repaired argument verifies both central winning statements.

Theorem 3.4Correct

The uniform-exponent bound uses the correct rational dimension after an index repair

Page 12 · proof of Theorem 3.4 · arXiv:2107.08084v1

If ξ∈L\xi\in\mathcal L had dim⁡Q(ξ)≤d−n\dim_{\mathbb Q}(\xi)\leq d-n, the rational subspace generated by ξ\xi could be extended to a rational (d−n)(d-n)-plane meeting the completely irrational nn-plane L\mathcal L nontrivially. Hence dim⁡Q(ξ)≥d−n+1.\dim_{\mathbb Q}(\xi)\geq d-n+1. When d−n≥2d-n\geq2, Proposition 3.3 applied at the actual rational dimension, together with the monotonicity Gr≥Gd−n+1G_r\geq G_{d-n+1} for r≥d−n+1r\geq d-n+1, supplies the required Gd−n+1G_{d-n+1} bound; its polynomial is exactly the degree-(d−n)(d-n) polynomial printed in the next display. Combining this with ω(ξ)≤n/(d−n)\omega(\xi)\leq n/(d-n) gives the polynomial defining Wn,d\mathfrak W_{n,d}. When d−n=1d-n=1, the asserted bound is Wn,d=1\mathfrak W_{n,d}=1 and follows directly from the standard bound ω^(ξ)≤1\hat\omega(\xi)\leq1 quoted on the same page, so Proposition 3.3 is not needed. Thus the theorem is correct; the two occurrences of d−nd-n before the polynomial are off-by-one typos in the nontrivial codimension-≥2\geq2 case.

Theorem 3.5Correct

Best approximations cannot eventually lie in too small a subspace

Page 13 · Theorem 3.5 and proof · arXiv:2107.08084v1

The span of all sufficiently late integer best-approximation vectors is a rational subspace. If its dimension were at most d−nd-n, it could be extended to a rational (d−n)(d-n)-plane. Complete irrationality gives a positive angle between this plane and L\mathcal L, while the defining approximation errors force the angles of the best-approximation vectors to L\mathcal L to tend to zero. This contradiction proves R(Θ)≥d−n+1R(\Theta)\geq d-n+1.

02Proofs3 reported findingsContains incorrect or incomplete proofs

The exponent and best-approximation arguments are correct after two off-by-one notation repairs. The algebraic-manifold escaping proof is incorrect as printed because its Taylor identity has wrong coefficients and it attempts to escape zero sets of identically zero derivatives. A verified Taylor-expansion repair establishes the needed estimate. The remaining set-operation, player-name, and index mismatches have unique harmless corrections.

Algebraic manifold escaping lemmaIncorrect as written · verified repair

The displayed Taylor identity is false, but the escaping estimate has a verified repair

Pages 6–8 · proof of the algebraic manifold escaping lemma, especially the display on page 7 · arXiv:2107.08084v1

The degree-ss Taylor expansion is printed with coefficient 1/k1/k in the ordered kk-fold derivative sum; the correct coefficient is 1/k!1/k!. The induction also asks to escape every zero set of ∂f/∂zi\partial f/\partial z_i, although some of those derivatives may be identically zero. Restrict the induction to the nonzero first derivatives and choose one whose absolute value is bounded below on the later ball. With the correct finite expansion, f(z)=∇f(a)⋅(z−a)+∑k=2s1k!∑i1,…,ik∂i1⋯∂ikf(a)∏j=1k(zij−aij),f(z)=\nabla f(a)\cdot(z-a)+\sum_{k=2}^{s}\frac1{k!}\sum_{i_1,\ldots,i_k}\partial_{i_1}\cdots\partial_{i_k}f(a)\prod_{j=1}^{k}(z_{i_j}-a_{i_j}), the remainder is at most K∑k=2srkδk/k!K\sum_{k=2}^{s}r^k\delta^k/k!. Choosing δ\delta so this is smaller than the displayed linear separation and then applying the hyperplane escaping lemma gives a later ball disjoint from {f=0}\{f=0\}. This repair applies identically at every downstream use.

Proof of Theorem 3.4Typo

Two adjacent rational-dimension indices are off by one

Page 12 · first paragraph of the proof of Theorem 3.4 · arXiv:2107.08084v1

The proof prints dim⁡Q(ξ)≥d−n\dim_{\mathbb Q}(\xi)\geq d-n and then invokes Gd−nG_{d-n}. Complete irrationality gives dim⁡Q(ξ)≥d−n+1\dim_{\mathbb Q}(\xi)\geq d-n+1. For d−n≥2d-n\geq2, Proposition 3.3 at the actual rational dimension and monotonicity in the index give the needed Gd−n+1G_{d-n+1} bound; the polynomial in the immediately following display already has the corresponding exponents. For d−n=1d-n=1, Proposition 3.3 does not apply because its index hypothesis is r≥3r\geq3, but the theorem reduces to the standard ω^≤1\hat\omega\leq1 bound. With that endpoint qualification, the two printed indices are harmless.

Winning-set constructionTypo

Union, player, and ambient-index notation have unique corrections

Pages 3–9 · Schmidt-game definitions, escaping lemmas, and proof of Theorem 2.1 · arXiv:2107.08084v1

The proof identifies complete irrationality with the complement of an intersection of bad algebraic sets; the required set is the complement of their union. The escaping strategy is repeatedly assigned to Bob although, under the paper's own game convention, it is Alice who enforces membership in a winning target. Finally, the Taylor bound uses dkd^k where the polynomial has rr variables. Replacing ⋂MSM\bigcap_{\mathcal M}S_{\mathcal M} by ⋃MSM\bigcup_{\mathcal M}S_{\mathcal M}, `Bob' by `Alice' in the escaping statements, and dkd^k by rkr^k gives the unique notation consistent with the surrounding construction and changes no argument.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper
arXiv:2107.08084v1
Authors listed
Vasiliy Neckrasov
Audit date
August 19, 2026
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