arXiv:2301.08212v2
On Furstenberg's Diophantine result
math.NT11K31
Abstract
We give a very simple and explicit exposition of the effective results on by Bourgain, Lindenstrauss, Michel and Venkatesh.
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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.
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Detailed mathematical audit
01Statements3 reported findingsCorrect
The effective density theorem for the coprime semigroup and its inhomogeneous irrational corollary are correct. The displayed local sign and rounding slips do not change the claims.
Theorems 1–2Correct
The semigroup orbit has the stated triple-logarithmic density
Pages 2–3 and 9–11 · Theorems 1–2 and final proof · arXiv:2301.08212v2
The combinatorial covering lemma and the Fourier estimate create a long enough progression in the semigroup orbit. Dirichlet approximation then transfers the rational statement to infinitely many irrational approximants with an error smaller than the target density scale.
Full paper, version 2 ↗Lemmas 1, 4, and 8Minor formal corrections · no status impact
Three ordering, rounding, and sign mismatches are local repairs
Pages 4, 6, and 9–10 · Lemmas 1, 4, and 8 · arXiv:2301.08212v2
The wrapped difference in Lemma 1 needs the opposite ordered pair in one branch; the block count in Lemma 4 must use a floor rather than a ceiling to preserve the lower size bound; and Lemma 8 requires in accordance with its defining correspondence. Each replacement preserves the asserted set membership and estimates.
Final paragraphMinor formal correction · no status impact
The modular orbit must be read through fractional parts
Page 10 · conclusion of Theorem 1 · arXiv:2301.08212v2
The proof writes before reduction modulo one. The required identity is for its fractional part, or equivalently for distance modulo one. Making that reduction explicit gives the announced density statement.
02Proofs1 reported findingCorrect
The elementary combinatorial-Fourier proof is complete after the local branch, floor, sign, and fractional-part corrections.
Sections 2–6Correct and complete after the stated repairs
The progression and Fourier bounds combine with compatible scales
Pages 3–11 · lemmas and proof of Theorem 1 · arXiv:2301.08212v2
The corrected block sizes retain a positive proportion of residues, the exponential-sum estimate is applied on exactly those blocks, and the final modular reduction converts the resulting approximation into density on the circle. The irrational perturbation in Theorem 2 can be chosen strictly smaller than the available margin.
03Novelty0 reported findingsNo non-novelty findings
No non-novelty findings.