MathAudit

Paper metadata

Original submission
August 6, 2023
Claimed MSC
arXiv categories
math.NT
Journal reference
Comb. Number Th. 13 (2024) 239-264

Abstract

Let v→∈R2∖Q2\overrightarrow{v}\in\mathbb{R}^2\setminus\mathbb{Q}^2, let ∥⋅∥\lVert\cdot\lVert be an arbitrary norm on R2\mathbb{R}^2, and let (qn,pn→)n=0∞⊂N×Z2(q_n,\overrightarrow{p_n})_{n=0}^{\infty} \subset\mathbb{N}\times\mathbb{Z}^{2} be the best approximation vectors sequence of v→\overrightarrow{v} with respect to ∥⋅∥\lVert\cdot\lVert. We define the nth long displacement vector of v→\overrightarrow{v} to be βn→:=qn+1(qnv→−pn→)\overrightarrow{\beta_n}:=\sqrt{q_{n+1}}(q_{n}\overrightarrow{v}-\overrightarrow{p_n}) and prove the existence of long displacement vectors who have non-typical properties, focusing on their length, direction, and congruence class.

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 19, 2026
01Statements3 reported findingsCorrect

The prescribed-length, direction, and congruence construction for long displacement vectors in dimension two, together with the stated limiting-spectrum and lattice-flow consequences, is correct. The detected interval-sequence and labeling defects do not change any mathematical conclusion.

Theorem 1Correct

Simultaneous control of long displacement vectors

Pages 5–6 and 10–35 · Theorem 1, Theorem 2, and the inductive construction · arXiv:2308.03049v1

For an arbitrary norm on R2\mathbb R^2, the construction chooses successive primitive triples {wn−2,wn−1,wn}\{\mathbf w_{n-2},\mathbf w_{n-1},\mathbf w_n\} and keeps the associated approximation cylinders free of interior lattice points. Proposition 3 then identifies the limiting sequence as the best-approximation sequence of v=lim⁡npn/qn\mathbf v=\lim_n\mathbf p_n/q_n. Property (2.4) gives the prescribed values of qn+1 ∥qnv−pn∥\sqrt{q_{n+1}}\,\lVert q_n\mathbf v-\mathbf p_n\rVert, property (2.6) gives the cone and alternating-quadrant conditions, and property (2.7) gives the requested residue class. The coefficient of wn−3\mathbf w_{n-3} is ±1\pm1, so primitivity is preserved. Since qn>nqn−1q_n>nq_{n-1}, the resulting vectors are not badly approximable. These implications establish all parts of the theorem.

Corollaries 1 and 2Correct

Interval in the limiting Dirichlet spectrum and uncountably many convergent vectors

Pages 6–7 · Corollaries 1 and 2 · arXiv:2308.03049v1

Apply Theorem 1 with a sequence of non-degenerate closed intervals shrinking to any prescribed c∈[0,M∥⋅∥]c\in[0,M_{\lVert\cdot\rVert}]. The resulting long-displacement norms converge to cc, and the theorem supplies continuum many distinct best-approximation sequences. The eventual uniqueness of the best-approximation sequence for a fixed vector prevents this continuum from collapsing to countably many vectors. Thus [0,M∥⋅∥][0,M_{\lVert\cdot\rVert}] lies in the limiting spectrum, and the set of Dirichlet-converging vectors is uncountable. For a diagonal ellipsoidal norm, the displayed value of MM agrees with the sharp Dirichlet constant, so the equality asserted in Corollary 1 follows.

Corollary 3Correct

Prescribed eventual height for the associated lattice trajectory

Pages 7–8 · Corollary 3 and its proof · arXiv:2308.03049v1

For Lv=(I2−v01)Z3L_{\mathbf v}=\begin{pmatrix}I_2&-\mathbf v\\0&1\end{pmatrix}\mathbb Z^3, the local maxima of the first minimum along gtLvg_tL_{\mathbf v} satisfy λ1(gtn+1Lv)3=qn+1∥qnv−pn∥2.\lambda_1(g_{t_{n+1}}L_{\mathbf v})^3=q_{n+1}\lVert q_n\mathbf v-\mathbf p_n\rVert^2. Choosing the long-displacement norms to converge to r3/2r^{3/2} therefore makes these local maxima converge to rr, proving the claimed eventual compact-exhaustion threshold. The invalid displayed interval sequence has the harmless replacement recorded under Proofs.

02Proofs5 reported findingsCorrect

The reduction to an inductively constructed primitive lattice basis and the geometric cylinder-avoidance argument are correct and complete at the level required for the central results. Two local interval-sequence or labeling defects admit verified corrections and do not affect the proof status.

Proposition 3 and the limit argumentCorrect and complete

Cylinder exclusion identifies the limiting best approximations

Pages 9 and 11–12 · Proposition 3 and proof of Theorem 1 from Theorem 2 · arXiv:2308.03049v1

The strict growth of qnq_n, strict decrease of the errors, and absence of an integer point in the interior of each approximation cylinder are exactly the defining minimality conditions for successive best approximations. If an integer pair contradicted the limiting condition, continuity would place it in the interior of the corresponding finite-stage cylinder for all sufficiently large construction stages, contradicting property (2.1). Passing properties (2.4) and (2.6) to the limit is valid because the prescribed intervals and cone are closed at the final stage.

Lemmas 1–10Correct and complete

Normalization and lattice-point selection preserve the induction

Pages 13–29 · Sections 3 and 4 · arXiv:2308.03049v1

The map GG fixes the denominator coordinate, sends the current primitive plane and its neighboring translate to horizontal planes, and preserves the ellipsoidal norm in the transverse coordinates. Its covolume relation qn−1L=H−1q_{n-1}L=H^{-1} follows from unimodularity. The regions B1B_1 and B2B_2 are arranged so that the relevant ellipsoidal cylinders contain no unintended points of the two adjacent lattice planes. Irrationality of the selected vertical spacing supplies infinitely many lattice candidates with every residue adjustment r mod mr\bmod m, while their denominators tend to infinity. These facts provide precisely the candidates used in the final induction.

Final inductive verificationCorrect and complete

Properties (2.1)–(2.7) close simultaneously

Pages 29–35 · Section 5 · arXiv:2308.03049v1

Increasing the candidate index makes qnq_n arbitrarily large and vn−vn−1\mathbf v_n-\mathbf v_{n-1} arbitrarily small, preserving all earlier strict inequalities. The direction of the new error approaches the negative of the preceding error, giving the alternating-quadrant condition. Lemma 10 fixes the limiting value of qn∥qn−1vn−pn−1∥2q_n\lVert q_{n-1}\mathbf v_n-\mathbf p_{n-1}\rVert^2 inside the target interval. The adjacent-plane cylinder exclusion proves (2.1), and choosing rr among a complete residue block proves (2.7). The ±1\pm1 coefficient on the third basis vector proves the next triple remains a lattice basis.

Proof of Corollary 3Minor formal correction

The displayed interval sequence needs an arbitrary finite prefix

Pages 7–8 · first paragraph of the proof of Corollary 3 · arXiv:2308.03049v1

The proof displays the target intervals as [r−1/(n+1),r−1/(n+2)]3/2[r-1/(n+1),r-1/(n+2)]^{3/2} for every n≥0n\ge0. For r=0r=0, and for finitely many initial nn whenever rr is small, these endpoints are negative and the 3/23/2 power is not a real interval. Choose arbitrary admissible non-degenerate intervals for those finitely many indices and use the displayed intervals once both endpoints are nonnegative (for r=0r=0, use any positive intervals shrinking to 00). The limiting local-maximum calculation is unchanged, so no statement or downstream argument requires correction.

Table following Theorem 1Typo

The table is attached to an empty numbered equation

Page 6 · empty display labeled (4) immediately before the table · arXiv:2308.03049v1

The source creates an empty equation solely to obtain label (4), then later refers to the table as equation (4). Attach the label to the table or call it “Table 1.” The four displayed constants themselves are obtained by substitution in formula (3), so this labeling defect has no mathematical effect.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper
arXiv:2308.03049v1
Authors listed
Alon Agin
Audit date
August 19, 2026
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