Published paper

Applications of Siegel's Lemma to a system of linear forms and its minimal points

Johannes Schleischitz

Published version ↗Open DOI ↗arXiv record →

Publication metadata

Journal citation: Combinatorics and Number Theory 11(2), 125–148 (2022)
Publication date: 2022
DOI: 10.2140/moscow.2022.11.125 ↗
arXiv counterpart: arXiv:1904.06121v4

Abstract

The paper develops lower bounds for ratios of Diophantine exponents and criteria for linear independence of minimal points; the focal paper cites its arXiv version for the rational-dimension form of the ratio bound.

Role in dependence graphs

Proof-critical source

On Some Properties of Irrational Subspaces

This paper is included only for the following marked statement:

Theorem 2.4, equation (6), rational-dimension specialization · arXiv:1904.06121v4, pp. 5–6Records the ratio bound in the dimension-sensitive form used for vectors that are not necessarily totally irrational.

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arXiv:1904.06121v4 · explicit fallback for inaccessible version of record

Johannes Schleischitz. Applications of Siegel's Lemma to a system of linear forms and its minimal points. arXiv:1904.06121v4.

The Combinatorics and Number Theory version of record was available only by subscription or purchase and could not be retrieved in this audit session; the exact arXiv v4 manuscript was reviewed and is not represented as the version of record.

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Generated August 23, 2026

01Statements5 reported findingsContains unsupported statements

The attributed one-form ratio theorem, the new short-vector bounds, and the linear-independence criteria are supported after local notation corrections. The generic form of Theorem 2.4 is recoverable from a corrected determinant estimate, and its exact one-dimensional specialization used by the focal paper is independently supported; the theorem's stated reduction for arbitrary $h$ is not verified because the manuscript gives only one sentence for that substantive step.

Theorem 2.1Correctly treated as an attributed result

The Marnat-Moshchevitin ratio bound is accurately recorded

arXiv:1904.06121v4, p. 3, Theorem 2.1

For one linear form, the stated lower bound $w(\Theta)/\widehat w(\Theta)\geq G_{1,n}$ and its defining polynomial agree with the independently audited Marnat-Moshchevitin theorem. The single occurrence of $\widehat\omega$ in the definition of $G_{1,n}$ is a notation slip for the paper's $\widehat w$ .

Theorem 2.4, equation (6)Not able to verify in its full stated generality

The arbitrary-subspace reduction is not established in the manuscript

arXiv:1904.06121v4, pp. 5-6 and 27, Theorem 2.4, equation (6), and Annex

After correcting the determinant factors, the Annex proves the generic case $h=m+n$ . For general $h$ , however, it only says that one can reduce to an $m'\times n'$ matrix by considering $\mathscr L_\Theta\cap\mathscr S_\Theta$ . It does not construct the rational lattice coordinates, prove equivalence of the restricted approximation norms and exponents, or show that the needed consecutive-minimal-point property survives. No complete repair of that general reduction was verified.

Theorem 2.4, marked specializationCorrect for the imported specialization

The rational-dimension ratio estimate used by the focal paper is correct

arXiv:1904.06121v4, pp. 5-6, Theorem 2.4 and equation (6), with $n'=1$ ; focal Proposition 3.3

For a vector whose coordinates have rational dimension $r\geq3$ , a rational change of coordinates reduces to a totally irrational vector in dimension $r-1$ without changing the ordinary or uniform simultaneous exponents. The audited Marnat-Moshchevitin one-form theorem then gives exactly $\omega/\widehat\omega\geq G_r$ . Thus the statement imported by the focal Irrational Subspaces paper remains supported even though the Annex does not prove the whole arbitrary- $h$ formulation. No external paper beyond Marnat-Moshchevitin is materially needed for this exact marked claim.

Theorems 3.1 and 3.3Correct after a local correction

The short-vector bounds follow, with one reciprocal typo in an equality clause

arXiv:1904.06121v4, pp. 7-9 and 18-21, Theorems 3.1 and 3.3 and proofs

Siegel's Lemma applied to a short lattice vector and a maximal independent subfamily yields the announced bounds in terms of $1+\tau+\cdots+\tau^{h-2}$ and the $h=3$ specialization. In the last equality clause of Theorem 3.3, the printed limit $\widehat w/w$ must be $w/\widehat w$ : the logarithmic height ratio is at least one, and the preceding inequalities force the latter quotient.

Theorems 4.1-4.6Correct after local corrections

The linear-independence criteria are supported after mechanical formula repairs

arXiv:1904.06121v4, pp. 10-17 and 20-27, Section 4 and proofs

The determinant relation among a maximal independent subfamily, Siegel's Lemma, and the growth and error quotients give the stated independence thresholds. In equation (39), the displayed comparison of the exact and simplified thresholds has the wrong direction: the simplified sufficient threshold is the smaller one and must appear first. The main condition using the exact threshold remains correct. The extra parenthesis in Theorem 4.6 and a few missing or doubled local symbols are uniquely repairable and do not change the criteria.

02Proofs4 reported findingsContains incorrect or incomplete proofs

The Siegel-Lemma arguments in Sections 3-5 are substantively correct after local notation repairs. The Annex proof of Theorem 2.4 is incorrect as printed because it swaps the numbers of height and error factors, and its extension from the generic case to arbitrary $h$ is only asserted.

Proof of Theorem 3.1Correct and complete after notation correction

The short-vector determinant argument has the required exponent balance

arXiv:1904.06121v4, pp. 18-21, Section 5.3

A maximal independent subfamily together with the prescribed short vector gives an integer relation whose primitive coefficient vector is bounded by Siegel's Lemma. Pairing that relation with the extended matrix controls the maximal coefficient from below; substitution of the uniform error estimate yields equation (48), and Lemma 5.1 turns the intervening growth ratios into the claimed geometric sum. The isolated $a_t$ in the linear identity must be $a_1$ , as the adjacent indexing makes clear.

Proofs of Theorems 4.1-4.6Correct and complete after notation correction

The consecutive-minimal-point criteria close after local symbol repairs

arXiv:1904.06121v4, pp. 20-27, Sections 5.4-5.5

Assuming dependence, a maximal independent subfamily supplies a primitive integer relation. The coefficient bound and the separation between consecutive errors force inequalities incompatible with each theorem's hypothesis. The proof's doubled exponent in the Theorem 4.2 estimate, a missing floor bracket in Theorem 4.3, and the unmatched parenthesis in Theorem 4.6 are mechanical: the following algebra already uses the uniquely corrected forms.

Annex determinant estimateIncorrect as written · verified repair for the generic case

The numbers of height and error columns are interchanged

arXiv:1904.06121v4, p. 27, first determinant estimate in the Annex

There are $n$ coordinate columns bounded by heights and $m$ transformed $y$ -columns bounded by errors. The printed product instead contains $m$ height factors and $n$ error factors. It must be $\prod_{j=m}^{m+n-1}\lVert z_{t+j}\rVert\prod_{j=0}^{m-1}\lVert\Theta^E z_{t+j}\rVert,$ not the product starting the height block at $t+n$ . With this swap, assuming every consecutive growth ratio is below $G$ gives the coefficient $-\widehat w\sum_{j=1}^{m-1}G^{-j}+1-\widehat w+\sum_{j=1}^{n-1}G^j=P_{m,n}(G)$ , so the generic conclusion follows. The printed range $j\in\{t+1,\ldots,t+n-2\}$ must likewise range over the eligible consecutive ratios in the full block; as printed it is empty when $n=2$ .

General-$h$ reduction in the AnnexIncomplete as written · no verified full repair supplied

The passage to the restricted system is only asserted

arXiv:1904.06121v4, p. 27, final paragraph of the Annex

The one-sentence instruction to consider $\mathscr L_\Theta\cap\mathscr S_\Theta$ omits the lattice-coordinate construction and the invariance statements needed to replace the original problem by an $m'\times n'$ minimal-point system. Those obligations are not merely a change of notation, so the full form of Theorem 2.4 is not proved here. For the exact $n'=1$ claim imported by the focal paper, the independent rational-coordinate reduction followed by the audited Marnat-Moshchevitin theorem supplies the needed result.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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