arXiv:2212.14646v1
On Korobov bound concerning Zaremba's conjecture
math.NTmath.CO11J7011B3005C2520G4011B75
Abstract
We prove in particular that for any sufficiently large prime there is such that all partial quotients of are bounded by . For composite denominators a similar result is obtained. This improves the well--known Korobov bound concerning Zaremba's conjecture from the theory of continued fractions.
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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 improvement of Korobov's bound to in the stated prime-factor, square-free, and prime-power regimes is correct.
Section 3Correct and complete
The square-free and repeated-prime regimes reduce to verified growth inputs
Pages 8–20 · square-free, prime-power, and general-modulus cases · arXiv:2212.14646v1
For square-free moduli, a determinant escape argument reduces concentration modulo every prime divisor to the prime case. For a large prime power, the congruence-level product-growth theorem reaches the required modular neighborhood. The final large-prime-factor case combines escape from the trace varieties with the pivot alternative, and the normalizer count supplies a torus satisfying the remaining intersection bound.
Theorem 1Correct
The bounded-partial-quotient numerator exists in each announced denominator regime
Pages 2–3 and 7–20 · Theorem 1 and Sections 2–3 · arXiv:2212.14646v1
The continued-fraction interval decomposition produces many candidate numerators. Expansion and escape estimates force the associated matrix product set to grow unless it already covers enough of the modular group; in either case one obtains a congruence solution avoiding the hyperbola obstruction. Lemma 2 then converts that avoidance into the asserted partial-quotient bound.
Full paper, version 1 ↗Appendix theoremCorrect
The large-deviation input has the required uniform dependence
Pages 20–25 · Theorem 14 and appendix · arXiv:2212.14646v1
The continuant is decomposed into logarithmic increments, the exceptional short increments are counted directly, and the exponential moment estimate is uniform in the growing digit bound. This supplies the input used in the general-modulus argument.
02Proofs1 reported findingCorrect
The prime, square-free, prime-power, and large-prime-factor arguments are complete; the expansion, escape, and continued-fraction inputs are applied within their stated regimes.
Sections 2–4Correct and complete
The product-growth dichotomy closes in every case
Pages 3–25 · Lemmas 2–13 and appendix · arXiv:2212.14646v1
The prime case follows directly from the mixing estimate. For composite moduli, the escape lemma controls concentration on trace varieties, while the pivot and large-set alternatives both yield the required growth. The algebraic normalizer bound supplies the final torus choice, and the prime-power argument treats the remaining repeated-factor regime.
03Novelty0 reported findingsNo non-novelty findings
No non-novelty findings.