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arXiv:1802.03081v4

An optimal bound for the ratio between ordinary and uniform exponents of Diophantine approximation

Antoine Marnat, Nikolay Moshchevitin

math.NT
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Original submission
February 8, 2018
Latest revision
April 1, 2020 · version 4
Claimed MSC
arXiv categories
math.NT

Abstract

We provide a lower bound for the ratio between the ordinary and uniform exponent of both simultaneous Diophantine approximation and Diophantine approximation by linear forms in any dimension. This lower bound was conjectured by Schmidt and Summerer and already shown in dimension 22 and 33. This lower bound is reached at regular systems presented in the context of parametric geometry of numbers, and thus optimal.

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Audited against arXiv v4

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

Generated August 20, 2026
01Statements2 reported findingsCorrect

The optimal lower bounds for ordinary-to-uniform exponent ratios, in both simultaneous and dual approximation, and the realization of every larger ratio are supported by the determinant and parametric-geometry arguments.

Theorem 1, lower boundsCorrect

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

Pages 4-5 and Sections 2-5 · first part of Theorem 1 · arXiv:1802.03081v4

The extracted best-approximation pattern spans the full rational space and decomposes recursively into overlapping lower-dimensional patterns. Repeated Schmidt height inequalities force at least one adjacent height or error to improve by the positive root of Rn,λ^R_{n,\widehat\lambda} in the simultaneous case. Hyperbolic rotation gives the dual argument and the root of Rn,ω^∗R^*_{n,\widehat\omega}.

Theorem 1, optimality and realizationCorrect

Every admissible larger ratio is realized

Section 6 · second part of Theorem 1 and Roy-system construction · arXiv:1802.03081v4

The three-parameter Roy systems have explicitly computed lower and upper Schmidt-Summerer exponents. Choosing their parameters fixes the prescribed uniform exponent and any ratio CC at or above the root bound. Roy's realization theorem then supplies infinitely many vectors with exactly those Diophantine exponents.

02Proofs4 reported findingsCorrect

The long induction tracks rational subspace heights and all positivity constraints consistently; the dual reduction and sharpness construction complete the two directions independently.

Lemma 4Correct

The local growth alternative has the required exponent balance

Section 2.3 · Lemma 4 and formulas (22)-(24) · arXiv:1802.03081v4

The determinant lower bound for the selected independent best approximations is combined with the uniform error estimate. With the listed weights, cancellation of the auxiliary scale leaves precisely the alternatives that either increase the next height by gg or decrease the current error by the corresponding uniform-exponent multiple.

Lemmas 5-8Correct and complete

The recursive best-approximation pattern exists in every dimension

Sections 3-4 · Lemmas 5-8 · arXiv:1802.03081v4

Starting beyond any index, the lemmas select consecutive independent blocks whose adjacent overlaps are complete rational lattices. The induction joins two dimension-nn patterns along their common subspace and retains enough endpoint vectors to span the full dimension, exactly the configuration needed for the subsequent height product.

Sections 4-5Correct

Schmidt height inequalities yield both lower bounds

Sections 4-5 · equations (46)-(85) and Lemma 10 · arXiv:1802.03081v4

The recursive weights are positive by Propositions 3-4 and sum so that every auxiliary power cancels. Hence one factor in the height product must dominate, giving the defining polynomial inequality. Lemma 10 transports the same rational-subspace determinant estimates through the hyperbolic rotation for one-linear-form approximation.

Section 6Correct

The Roy-system sharpness argument computes the claimed exact exponents

Section 6 · Roy systems and final proof of Theorem 1 · arXiv:1802.03081v4

The slopes and switch points satisfy the generalized-system axioms. Direct liminf and limsup calculations give the target ordinary and uniform exponents, including the boundary root case, and the cited realization theorem converts each system into a vector with rationally independent coordinates.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper
arXiv:1802.03081v4
Authors listed
Antoine Marnat, Nikolay Moshchevitin
Audit date
August 20, 2026
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