8 pages, correction of misprints, submitted to Czechoslovak Mathematical Journal
Abstract
Let Θ=(θ1,θ2,θ3)∈R3. Suppose that 1,θ1,θ2,θ3 are linearly independent over Z. For Diophantine exponents α(Θ)=sup{γ>0:t→+∞limsuptγψΘ(t)<+∞},β(Θ)=sup{γ>0:t→+∞liminftγψΘ(t)<+∞} we prove β(Θ)≥1/2(α(Θ)/1−α(Θ)+α(Θ)/1−α(Θ))2+4α(Θ)/1−α(Θ))α(Θ)
AI-generated audit
Audit summary
Audited against arXiv v2
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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.
Current report
Detailed mathematical audit
Generated August 20, 2026
01Statements2 reported findings✓Correct⌄
The lower bound for the ordinary exponent is correct. The endpoint α(Θ)=1 needs one local domain clarification but changes no mathematical conclusion.
Theorem 5✓Correct
Three-dimensional exponent bound
Page 3 · Theorem 5 · arXiv:1009.0987v2
For 1/3≤α(Θ)<1, the best-approximation argument proves β(Θ)≥α(Θ)g3(α(Θ)). The defining parameter b satisfies b2+1−ααb−1−αα=0, which is exactly the identity needed in both determinant alternatives.
Theorem 5 endpoint!Minor formal correction
The case α(Θ)=1 should be stated separately
Pages 2–3 · definition of g3 and Theorem 5 · arXiv:1009.0987v2
The function g3 is defined only on [1/3,1), whereas Theorem 5 does not exclude α(Θ)=1. State the displayed formula for α(Θ)<1 and add that α(Θ)=1 gives β(Θ)=+∞, or extend the right-hand side by its value +∞. The endpoint conclusion follows by applying the already proved bound to every α<1 and letting α↑1. This is a verified local domain repair and affects no later result.
02Proofs3 reported findings✓Correct⌄
The two- and three-dimensional lattice estimates and the final alternative argument are correct and complete.
Lemma 1✓Correct and complete
Covolume estimate in a two-dimensional block
Pages 4–5 · Lemma 1 and Equations (6)–(10) · arXiv:1009.0987v2
The section of the best-approximation cylinder has two independent boundary lattice points and no nonzero interior lattice point. The lower area estimate and Minkowski's theorem therefore give the claimed two-sided comparison ζlxl+1≍ΘdetΛ.
Lemma 2✓Correct and complete
Three-dimensional projection determinant
Pages 5–6 · Lemma 2 and Equations (11)–(15) · arXiv:1009.0987v2
The cofactor vector is orthogonal to the three-dimensional rational subspace. Equations (14) and (15) force one of the three coordinate-projection determinants to be at least the stated fixed multiple of the simplex volume, and integrality connects that volume to the primitive lattice covolume.
Proof of Theorem 5✓Correct and complete
The two determinant alternatives yield the same exponent
Pages 6–7 · Equations (17)–(20) and proof of Theorem 5 · arXiv:1009.0987v2
The two primitive normals are independent in Λ⊥, so their norm product dominates detΛ⊥=detΛ. In the first alternative the exponent is (1−α)bα; in the second it is 1−αb+α(1−b). The quadratic identity for b makes both equal to g3(α), after which ζl≤xl+1−α supplies the claimed ordinary exponent.
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Paper
arXiv:1009.0987v2
Authors listed
Nikolay Moshchevitin
Audit date
August 20, 2026
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