arXiv:2103.12113v2
On transference principle and Nesterenko's linear independence criterion
math.NT
Abstract
We consider the problem of simultaneous approximation of real numbers with rationals and the dual problem of approximating zero with the values of the linear form at integer points. In this setting we analyse two transference inequalities obtained by Schmidt and Summerer. We present a rather simple geometric observation, which proves their result. We also derive several corollaries previously unknown. Particularly, we show that, together with the transference inequalities for uniform exponents, Schmidt and Summerer's inequalities imply the inequalities by Bugeaud and Laurent and "one half" of the inequalities by Marnat and Moshchevitin. Besides that, we show that our main construction provides a rather simple proof of Nesterenko's linear independence criterion.
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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.
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Detailed mathematical audit
01Statements3 reported findingsCorrect
The transference inequality and the application to Nesterenko's linear-independence criterion are correct. One signed-distance definition is printed as an unsigned distance and is treated as a notation typo.
Main transference theorem and Nesterenko corollaryCorrect
The exponent and dimension conclusions follow from the empty-cylinder construction
Pages 3 and 8–13 · main theorems and application · arXiv:2103.12113v2
Successive empty cylinders produce linearly independent lattice vectors with the required determinant control. Applying the dual convex-body estimate to the prescribed sequence of linear forms gives the announced lower bound for the rational dimension.
Full paper, version 2 ↗Empty-cylinder lemmaTypo · no status impact
The height coordinate must be signed on the chosen half-cylinder
Pages 8–9 · empty-cylinder lemma and its figure · arXiv:2103.12113v2
The text defines as an unsigned distance, making the cylinder centrally symmetric, but immediately uses on one half and later writes the union with its negative. Define as the oriented coordinate along the distinguished line, or define the displayed set as the positive half-cylinder. The figure and every application use that intended one-sided object.
Propositions 1–3Correct
The exponent transference consequences follow from the main inequality
Pages 5–8 · Propositions 1–3 and Corollaries 1–2 · arXiv:2103.12113v2
Substituting the ordinary and uniform simultaneous and dual exponents into the two cases of the empty-cylinder inequality gives the ratios in Proposition 1. Taking the appropriate maximum and minimum yields Proposition 2 and its corollaries, while the elementary comparison of the resulting factors proves Proposition 3. The parameter ranges used in each substitution match the definitions.
02Proofs1 reported findingCorrect
The proof is complete once the empty cylinder is read with its intended signed height; no estimate depends on the erroneous unsigned notation.
Sections 3–5Correct and complete after the stated repair
The one-sided cylinder and determinant estimates are compatible
Pages 8–13 · geometric construction and Nesterenko application · arXiv:2103.12113v2
With the signed height, the chosen half-cylinder contains no nonzero lattice point and its negative handles the opposite orientation. Minkowski's theorem, the determinant lower bound, and the limiting exponent calculation then apply exactly as written.
03Novelty0 reported findingsNo non-novelty findings
No non-novelty findings.