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Paper metadata

Original submission
November 22, 2016
Claimed MSC
11J04, 11J06
arXiv categories
math.NT
Comments
The paper is written in German. Submitted to the Proceedings of the conference "The Sixth International Conference Analytic and Probabilistic Methods in Number Theory", 11-17 September 2016, Palanga, Lithuania

Abstract

We study two irrationality measure functions ψα[2](t)ψ_α^{[2]} (t) and ψα[2]∗(t)ψ_α^{[2]*} (t) related to the "second best" approximations to a real numbers and prove some results on the structure of the corresponding Diophantine spectra. It happened that the first two elements of the spectrum for the function ψα[2]∗(t)ψ_α^{[2]*} (t) are associated with the numbers 1+52\frac{1+\sqrt{5}}{2} and e=∑n=0∞1n!e =\sum_{n=0}^\infty \frac{1}{n!}.

AI-generated audit

Audit summary

Audited against arXiv v1

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.

Current report

Detailed mathematical audit

Generated August 20, 2026
01Statements4 reported findingsCorrect

The continued-fraction classifications of the two second-best approximation functions and the resulting spectrum statements agree with the complete case analysis in the paper.

Satz 1Correct

The discontinuities of ψα[2]\psi_\alpha^{[2]} are correctly classified

Section 2 · Satz 1 and Hilfssatz 1-7 · arXiv:1611.07183v1

The five rules cover partial quotients at least three, equal to two, isolated ones, finite blocks of ones, and an infinite terminal block. The parallelogram lemmas enumerate all primitive non-convergent lattice points that can become minimizers and order their denominators exactly as listed.

Satz 2Correct

The reduced-fraction variant has the stated simpler rule

Section 3 · Satz 2 and Hilfssatz 8-12 · arXiv:1611.07183v1

After identifying scalar multiples of convergents through the rational-value exclusion, the remaining candidates in each continued-fraction cell are qn−2+jqn−1q_{n-2}+jq_{n-1} for 1≤j<an1\leq j<a_n, with the separate an=1a_n=1 candidate. The boundary cases in Bemerkung 2 account for the only duplicated initial denominator.

Satz 3Correct

The extremal and interval statements for L2\mathbb L_2 follow from the local formulas

Section 4.3 · Satz 3 and Hilfssatz 13 · arXiv:1611.07183v1

The formula expressing the second-approximation constant through adjacent continued-fraction tails reduces the upper edge to the classical Lagrange spectrum. The golden-ratio and 17\sqrt{17} equivalence classes give the stated maximum, gap, and isolated point, while the cited continued-fraction Cantor-sum result supplies the displayed initial interval.

Satz 4Correct

The reduced second-approximation spectrum is consistently derived

Section 4.4 · Satz 4 · arXiv:1611.07183v1

The explicit formulas from Satz 2 give maximum 5\sqrt5, the gap above 3/23/2, and minimum 1/21/2. The continued fractions for ee and the constructed tail families give the claimed accumulation and interval statements.

02Proofs2 reported findingsCorrect

The lattice-parallelogram enumeration and continued-fraction tail calculations cover the stated cases without leaving a substantive proof gap.

Haupthilfssatz 1 and Hilfssatz 1-7Correct and complete

The lattice-cell argument excludes all missing candidates

Section 2 · Haupthilfssatz 1 and Hilfssatz 1-7 · arXiv:1611.07183v1

Each pair of proposed consecutive discontinuities lies on the boundary of a determinant-one parallelogram. Writing any enclosed lattice point in the convergent basis shows it is either a listed boundary point, zero, or a convergent and hence excluded by definition. The seven lemmas match all local partial-quotient patterns.

Hilfssatz 13 and spectrum proofsCorrect

The comparison with the Lagrange constant is valid

Section 4.2-4.4 · Hilfssatz 13 and proofs of Satz 3-4 · arXiv:1611.07183v1

The products of candidate denominators and errors are rewritten using forward and backward continued-fraction tails. The resulting inequalities compare the second constants with twice the classical constant in the required regimes, and the equality cases are checked against the classical equivalence classification.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper
arXiv:1611.07183v1
Authors listed
Nikolay G. Moshchevitin
Audit date
August 20, 2026
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