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Publication metadata

Journal citation
Mathematika 66(3), 818–854 (2020)
Publication date
2020-06-04
arXiv counterpart
arXiv:1802.03081v4
Publication license
CC BY 4.0 ↗

Abstract

The paper establishes the sharp lower bound relating ordinary and uniform Diophantine exponents that is used to bound uniform exponents inside badly approximable subspaces.

Role in dependence graphs

Proof-critical source

On Some Properties of Irrational Subspaces

This paper is included only for the following marked statement:

  • Main ratio theorem · Theorem 1Gives the sharp lower bound for the ratio of ordinary to uniform Diophantine exponents.

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Mathematika 66(3) (2020), 818-854 (version of record)

Antoine Marnat and Nikolay G. Moshchevitin, "An optimal bound for the ratio between ordinary and uniform exponents of Diophantine approximation," Mathematika 66(3) (2020), 818-854.

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Generated August 23, 2026
01Statements2 reported findingsCorrect

The published article's optimal lower bounds for the ratios between ordinary and uniform Diophantine exponents, in both simultaneous and dual approximation, and its realization of every larger admissible ratio are supported by the arguments in the version of record.

Theorem 1, lower boundsCorrect

The ratios are bounded by the positive roots GG and G∗G^*

Mathematika 66 (2020), pp. 820-821, Theorem 1; proofs on pp. 825-849

For simultaneous approximation, the selected best-approximation pattern recursively decomposes into overlapping rational subspaces. Schmidt's height inequality and Lemma 4 force a growth alternative whose optimized exponent is the positive root of Rn,λ^R_{n,\widehat{\lambda}}. The hyperbolic rotation in Section 5 supplies the corresponding dual estimate governed by Rn,ω^∗R^*_{n,\widehat{\omega}}.

Theorem 1, realizationCorrect

Every admissible larger ratio is realized

Mathematika 66 (2020), pp. 849-853, Section 6 and the second part of Theorem 1

The self-similar Roy systems satisfy the generalized-system axioms and have explicitly computed liminf and limsup data. Choosing the three parameters fixes the prescribed uniform exponent and any ratio CC at or above the root bound; Roy's realization theorem then produces points with exactly those exponents.

02Proofs4 reported findingsCorrect

The local determinant estimate, the recursive rational-subspace construction, the dual reduction, and the Roy-system realization argument are mutually consistent and cover the claimed parameter ranges in the version of record.

Lemma 4Correct and complete

The local growth alternative has the required exponent balance

Mathematika 66 (2020), pp. 825-827, Lemma 4 and equations (16)-(24)

The determinant comparison splits into the two stated cases. Substitution of the uniform error estimate cancels the auxiliary powers exactly, and the positivity hypothesis gives the needed sign in the second case. Both branches yield the announced height growth exponent g(s,t)g(s,t).

Lemmas 5-8Correct and complete

The recursive best-approximation pattern exists in every dimension

Mathematika 66 (2020), pp. 828-840, Sections 3-4, Lemmas 5-8

The construction selects consecutive independent blocks and joins two lower-dimensional patterns along their common rational subspace. The recursive weights remain positive and their determinant exponents telescope as claimed, leaving precisely the polynomial recurrence that defines G(n,α)G(n,\alpha).

Dual reductionCorrect and complete

The hyperbolic rotation preserves the required height estimates

Mathematika 66 (2020), pp. 840-849, Section 5, especially Lemmas 9-10

The determinant estimates for the rotated lattices have the stated scale factors, which cancel from the Schmidt-height products. The resulting four alternatives reproduce the dual root equation and cover approximation by one linear form.

Roy-system constructionCorrect and complete

The sharpness construction computes the claimed exact exponents

Mathematika 66 (2020), pp. 849-853, Section 6

The component ordering, slopes, switch points, and self-similar endpoint matching satisfy the generalized-system conditions. Direct extrema of the first and last components give the requested ordinary and uniform exponents, including the boundary root case.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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