Published paper

On Some Properties of Irrational Subspaces

Vasiliy Neckrasov

MSC 2020

11J13

Published version ↗Published PDF ↗Open DOI ↗arXiv record →

Publication metadata

Journal citation: Uniform Distribution Theory 17(1), 89–104 (2022)
Publication date: 2022
DOI: 10.2478/udt-2022-0002 ↗
arXiv counterpart: arXiv:2107.08084v1
MSC 2020: 11J13
As specified in the publication (MSC 2020): 11J13 ↗Published PDF, first page
Publication license: CC BY-NC-ND 4.0 ↗

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 $\xi$ from a two-dimensional badly approximable completely irrational subspace of $\mathbb{R}^d$ one has $\widehat{\omega}(\xi)\leq\frac{\sqrt{5}-1}{2}$ . Besides that, some statements about the dimension of subspaces generated by best approximations to a completely irrational subspace easily follow from properties that we discuss.

Role in dependence graphs

Proof-critical source

On Some Properties of Irrational Subspaces

This paper is included only for the following marked statements:

Theorem 2.1 and Corollary 2.2 · Published pp. 94–98Winning completely irrational subspaces and the existence/full-dimension conclusion after intersection with badly approximable systems.
Theorem 2.3 · Published pp. 94 and 98Hyperplane-absolute-winning versions of the completely irrational and badly approximable completely irrational sets.
Theorem 3.4 · Published pp. 100–101The uniform-exponent upper bound for every vector in a badly approximable completely irrational subspace.
Corollary 3.7 · Published pp. 102–104For a good completely irrational two-dimensional subspace of $mathbb{R}^4$ , the eventual best approximations span all four dimensions.

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Version of record · Uniform Distribution Theory 17(1), 89–104 (2022)

Vasiliy Neckrasov. On Some Properties of Irrational Subspaces. Uniform Distribution Theory 17(1), 89–104 (2022).

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Generated August 23, 2026

01Statements4 reported findingsContains unsupported statements

The winning result for completely irrational matrices, its hyperplane-absolute-winning analogue, the exponent bound, and the best-approximation conclusions are correct after verified repairs to the printed arguments. The claims that the badly approximable completely irrational locus is winning and hyperplane absolute winning are not fully verified because their exact cited source chain could not be checked; no counterexample to either claim was found.

Theorems 2.1 and 2.3(1)Correct

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

Version of record, pp. 94–99 · Theorems 2.1 and 2.3(1), Lemmas 2.4–2.6, and proofs

For every rational complementary-dimensional subspace, nontrivial intersection with the graph of a matrix is the zero set of a nonzero determinant polynomial. The corrected algebraic-manifold arguments let Alice force a later game ball a positive distance from each such zero set. Enumerating the countably many rational subspaces and treating them successively preserves every earlier separation. This proves Theorem 2.1. The same induction in the hyperplane absolute game proves Theorem 2.3(1). The required corrections to the printed strategies are fully verified under Proofs.

Corollary 2.2 and Theorem 2.3(2)Not able to verify

The badly approximable intersections retain an unresolved exact-source obligation

Version of record, pp. 93–94 and 99 · Propositions 1.7–1.8, Corollary 2.2, and Theorem 2.3(2)

Corollary 2.2 claims that the badly approximable completely irrational locus is $\alpha$ -winning for every $\alpha\leq1/2$ , and Theorem 2.3(2) claims that this locus is HAW for every $n,m\in\mathbb N$ . Both conclusions intersect the independently repaired completely irrational locus with a cited winning set of badly approximable systems. The exact 1969 Journal of Number Theory article cited for Proposition 1.7 and the exact 2013 Journal of Number Theory article cited for Proposition 1.8 could not be retrieved. The available arXiv manuscript for the latter result materially imports two lemmas from the inaccessible 1969 article, so the proof-critical chain remains unverified. This is an evidentiary limitation, not evidence that either intersection statement is false.

Theorem 3.4Correct

The uniform-exponent bound follows after the rational-dimension index is corrected

Version of record, pp. 100–102 · Propositions 3.1–3.3, Theorem 3.4, and its example

For nonzero $\xi\in L$ , the least rational subspace containing $\xi$ has dimension $\dim_{\mathbb Q}(\xi)$ . If this dimension were at most $d-n$ , it could be extended to a rational $(d-n)$ -plane meeting the completely irrational $n$ -plane $L$ , a contradiction. Hence $\dim_{\mathbb Q}(\xi)\geq d-n+1$ . When $d-n\geq2$ , the ratio estimate at the actual rational dimension, its monotonicity in that dimension, and $\omega(\xi)\leq n/(d-n)$ yield the polynomial printed in the proof and the stated root bound. When $d-n=1$ , the defining root is $1$ and the result is the standard bound $\widehat\omega(\xi)\leq1$ . Thus the theorem is correct after the two adjacent index corrections recorded under Proofs.

Theorem 3.5 and Corollary 3.7Correct

Complete irrationality forces the asserted best-approximation span

Version of record, pp. 102–103 · Theorem 3.5, Proposition 3.6, and Corollary 3.7

If all sufficiently late integer best-approximation vectors lay in a subspace of dimension at most $d-n$ , their integer span could be extended to a rational $(d-n)$ -plane $B$ . Complete irrationality gives a positive angle between $B$ and $L$ , so $\operatorname{dist}(z_\nu,L)\geq D\lVert z_\nu\rVert\to\infty$ , contradicting the defining approximation errors. Therefore $R(\Theta)\geq d-n+1$ . In the two-by-two case, the independently checked source alternative $R(\Theta)\in\{2,4\}$ then forces $R(\Theta)=4$ .

02Proofs6 reported findingsContains incorrect or incomplete proofs

The printed Schmidt-game and algebraic-manifold arguments contain formally invalid steps, but complete repairs establish the self-contained winning conclusions. The hyperplane-absolute proof also needs a nonzero-polynomial hypothesis and a legal deletion width. The exponent and best-approximation arguments close after a rational-dimension index repair. The two badly approximable intersection claims remain unverified through their cited inputs.

Lemma 2.4 and proof of Theorem 2.1Incorrect as written · verified repair

The printed escaping strategy is assigned to the wrong player and parameter

Version of record, pp. 91–98 · Proposition 1.4, Lemma 2.4, and proof of Theorem 2.1

Under the paper's definition, Alice chooses the balls $A_i$ and must force a winning outcome. Proposition 1.4 and Lemma 2.4 instead start from an Alice ball, give Bob the forcing strategy, and use $\gamma=1+\alpha\beta-2\beta$ , while Theorem 2.1 assumes $2\alpha<1+\alpha\beta$ . A Bob strategy under the different condition does not prove Alice winning. Use the actual Alice escaping form: start from a Bob ball $B_k$ , put $\gamma'=1+\alpha\beta-2\alpha>0$ , and let Alice move maximally in the chosen direction until a later Bob ball lies in the required half-space. Schmidt's exact 1966 proof verifies this form. Replacing the affected $A$ -balls by $B$ -balls and $\gamma$ by $\gamma'$ throughout Lemma 2.4 preserves every radius and separation estimate and supplies precisely the strategy used in Theorem 2.1. Repair classification: Verified repair.

Algebraic-manifold inductionIncorrect as written · verified repair

Zero derivatives and a false absolute-value inference require correction

Version of record, pp. 95–96 and 98–99 · proofs of Lemmas 2.4 and 2.5, especially equations (5)–(8)

The induction demands $|\partial f/\partial z_i|>\varepsilon_{s-1}$ for every $i$ , which is impossible when a first derivative is identically zero. A nonzero constant polynomial has empty zero set and is immediate; for a nonconstant polynomial, apply the induction only to the nonzero first derivatives. At least one exists, and after the finite sequential escapes the gradient is bounded away from zero. The published version already has the correct Taylor coefficient $1/k!$ , but its remainder count $d^k$ must be $r^k$ , the number of ordered index choices for a polynomial in $r$ variables. Finally, equations (7)–(8) give one-sided scalar bounds, not bounds for both absolute values. Replace the last displayed comparison by $\lVert y-z\rVert\geq (u,y-z)=(u,y-o)-(u,z-o)>\alpha\beta\gamma'\delta/8$ . These corrections verify the escape estimate and apply equally to the HAW induction. Repair classification: Verified repair.

Lemma 2.5Incorrect as written · verified repair

The HAW lemma includes the zero polynomial and prints an illegal deletion width

Version of record, pp. 98–99 · Lemma 2.5 and its proof

Lemma 2.5 quantifies over every polynomial, but for $f\equiv0$ its zero set is all of $\mathbb R^r$ and no positive-distance conclusion is possible. Add the hypothesis that $f$ is nonzero; every determinant polynomial used downstream satisfies it. The proof then chooses deletion thickness $\beta\rho_t$ , although the game's rule requires it to be strictly less than $\beta\rho_t$ . Choosing, for example, thickness $3\beta\rho_t/4$ is legal and, since $\rho_t>\beta\delta/2$ , still exceeds the $\beta^2\delta/4$ strip containing the zero set. Together with the nonzero-derivative repair above, this completes the proof without changing Theorem 2.3. Repair classification: Verified repair.

Proof of Theorem 3.4Incomplete as written · verified repair

The rational-dimension step needs an off-by-one correction and endpoint argument

Version of record, p. 101 · first paragraph of the proof of Theorem 3.4

The proof prints $\dim_{\mathbb Q}(\xi)\geq d-n$ and then invokes $G_{d-n}$ . Complete irrationality gives $\dim_{\mathbb Q}(\xi)\geq d-n+1$ . For $d-n\geq2$ , apply Proposition 3.3 at the actual rational dimension $r$ and use $G_r\geq G_{d-n+1}$ for $r\geq d-n+1$ ; this monotonicity follows directly by substituting the root equation for $G_r$ into the next-index polynomial. The immediately following focal polynomial already has exactly the exponents obtained from $G_{d-n+1}$ . For $d-n=1$ , Proposition 3.3 is outside its $r\geq3$ range, but the defining root is $1$ and the theorem follows from the standard bound $\widehat\omega\leq1$ quoted on p. 99. Because the printed proof omits this monotonicity and endpoint treatment, the defect is substantive rather than a mere index typo. Repair classification: Verified repair.

Proof of Theorem 3.5Correct and complete

The angle argument covers every possible smaller eventual span

Version of record, pp. 102–103 · proof of Theorem 3.5 and deduction of Corollary 3.7

The eventual span of integer best approximations is rational. If its dimension is below $d-n$ , extending it to a rational $(d-n)$ -plane preserves containment of the tail. Compactness of the two disjoint unit-sphere sections gives a positive angular separation, while the best-approximation errors tend to zero and their norms tend to infinity. This contradiction proves the lower bound. The exact surveyed alternative for two-by-two matrices then yields Corollary 3.7.

Corollary 2.2 and Theorem 2.3(2)Not able to verify

The cited winning inputs could not be verified in their exact versions

Version of record, pp. 93–94 and 99 · Propositions 1.7–1.8 and their two intersection deductions

The intersection arguments themselves are immediate once the two cited permanence and badly approximable winning inputs are available. The exact Schmidt 1969 version and the exact Broderick-Fishman-Simmons journal version were inaccessible, and the available manuscript for the latter still depends materially on two unverified Schmidt lemmas. Repair classification: No repair supplied for this evidentiary obligation; an exact-source check or an independent proof of those inputs is needed.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

04Sources5 reported findingsContains incorrect or incomplete source use

Most proof-critical source uses are mathematically supported by exact versions or explicit fallbacks, but the Schmidt escaping input is invoked with the wrong controlling player and parameter. In addition, the exact Schmidt 1969, Broderick-Fishman-Simmons, and Kleinbock-Moshchevitin-Weiss journal versions could not all be checked. No missing proof-critical citation beyond the represented graph was identified.

Schmidt game input [8]Incorrect as invoked · verified repair

The escaping result is inapplicable in the form used by the focal proof

Version of record, pp. 91–98 · Propositions 1.2–1.4, Lemma 2.4, and proof of Theorem 2.1

The graph terminates this branch at Schmidt's 1980 book, whose exact text was not separately audited. The focal paper states the escaping step for Bob from an Alice ball with $1+\alpha\beta-2\beta>0$ , then uses it as though it supplied Alice's strategy under $1+\alpha\beta-2\alpha>0$ . Those are different source applications. Schmidt's exact 1966 version-of-record proof independently gives the needed White/Alice form from a Bob ball with the latter parameter, so the focal theorem is repairable, but the cited input is not correctly invoked as printed.

Schmidt 1966 version-of-record scan ↗

Schmidt 1969 and Broderick-Fishman-Simmons [2,9]Not able to verify

The badly approximable winning inputs remain unverified in their exact cited versions

Version of record, pp. 93–94 and 99 · Propositions 1.7–1.8, Corollary 2.2, and Theorem 2.3(2)

The exact 1969 and 2013 Journal of Number Theory versions were inaccessible after the recorded access checks. The exact arXiv v1 fallback for the 2013 paper does not close the obligation: its homogeneous HAW theorem uses two dimension lemmas imported from the inaccessible 1969 article. Thus the focal intersection deductions are formally applicable if those inputs hold, but the exact represented source chain could not be completed.

Broderick-Fishman-Simmons arXiv v1 fallback ↗

Broderick-Fishman-Kleinbock-Reich-Weiss [1]Correct and sufficient

The HAW permanence properties used by the focal paper are supported

Version of record, pp. 92 and 99 · Propositions 1.5–1.6 and proof of Theorem 2.3

The exact Cambridge version of record proves countable-intersection stability for hyperplane absolute winning sets and the implication from HAW to Schmidt winning in the required parameter range. These inputs are correctly used for the focal intersection and comparison statements.

Broderick-Fishman-Kleinbock-Reich-Weiss version of record ↗

Exponent sources [4,5,7]Supported by checked versions; exact KMW journal version unverified

The imported exponent inequalities are supported, with one exact-version limitation

Version of record, pp. 100–102 · Propositions 3.1–3.3 and proof of Theorem 3.4

The exact Marnat-Moshchevitin version of record proves the ordinary-to-uniform ratio bound. The exact Schleischitz arXiv v4 source has an incomplete proof of its full arbitrary-rational-dimension formulation, but the precise one-dimensional specialization imported here is independently obtained by rational coordinate reduction and the checked Marnat-Moshchevitin theorem. The exact Kleinbock-Moshchevitin-Weiss journal version was inaccessible; its exact arXiv v2 fallback correctly proves the ordinary-exponent bound used here. Thus the mathematical chain is supported, while equality with the inaccessible KMW version of record is not certified.

Marnat-Moshchevitin version of record ↗

Moshchevitin survey [6]Correct and sufficient

The exact best-approximation alternative used in Corollary 3.7 is supported

Version of record, p. 103 · Proposition 3.6 and Corollary 3.7

The exact Russian Mathematical Surveys version gives Corollary 4 to Theorem 7: for a good two-by-two matrix of full rational dimension, the eventual span dimension is either $2$ or $4$ . Complete irrationality and Theorem 3.5 exclude the first branch. The separate defect found in that survey's Theorem 68 is unrelated to this source use.

Moshchevitin survey version of record ↗

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