arXiv:2603.14116v2

On some results of Korobov and Larcher and Zaremba's conjecture

Ilya D. Shkredov

math.NTmath.CAmath.CO11J7011B3005C2520G4011B75

Paper metadata

Original submission: March 14, 2026
Latest revision: June 24, 2026 · version 2
Claimed MSC: 11J70, 11B30, 05C25, 20G40, 11B75
arXiv categories: math.NT, math.CA, math.CO
Paper license: CC BY 4.0 ↗
Comments: 41 pages. 1. Some minor typos have been corrected, and clarifications have been made. 2. For the reader's convenience, the proof of technical Lemma 32 has been broken down into three auxiliary lemmas, simplified, and clarified. 3. In the appendix, we provide a crude upper bound for the absolute constant $\mathcal{M}$ in the case of a prime denominator

Abstract

We prove, in particular, the well--known Zaremba conjecture from the theory of continued fractions for any prime denominator. More precisely, we show, firstly, that under some mild conditions, for any sufficiently large $q$ , there exists $a$ coprime to $q$ such that all partial quotients of $a/q$ are bounded by $O(\sqrt{\log q})$ , and, moreover we find asymptotically tight lower bound for the number of such $a$ . Secondly, we obtain a good lower bound for the number $a$ such that the sum of all partial quotients of $a/q$ is bounded by $O(\log q \cdot \sqrt{\log \log q})$ . This, accordingly, improves on some results of Korobov and Larcher. Finally, we show that for all sufficiently large $\mathcal{M}$ there are $Ω(q^{1-O(1/\mathcal{M})})$ numbers $a$ coprime to $q$ such that all partial quotients of $a/q$ are bounded by $\mathcal{M}$ .

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Audit summary

Audited against arXiv v2

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 15, 2026

01Statements3 reported findingsContains unsupported statements

No counterexample was found to the central conclusions. Theorems 6–7 are not able to be verified because their proof chain contains unresolved gaps even in the prime case, while their composite branches also use false counting inputs. The composite branch of Theorem 8 is likewise not verified. Corollary 1 is correct by a later complete proof.

Theorems 6–7Not able to verify

The full stated conclusions are not established by the printed proof

Pages 4–5 and 25–36 · Theorems 6–7, Lemma 32, Proposition 35, and Sections 5.2–5.3 · arXiv:2603.14116v2

The prime and composite arguments use the construction supplied by Lemma 32 and Proposition 35. Equation (107) in Lemma 32 drops an uncontrolled term, and Proposition 35 chooses coefficient bounds only after seeing the output of a lemma that requires those bounds as fixed inputs. No uniform or simultaneous selection argument repairs these obligations. The composite branches additionally depend on Lemmas 14–15 and Corollary 16, whose full-ring main terms are formally false. These defects prevent verification but do not provide a counterexample to either theorem.

Theorem 8Not able to verify

The stated composite-modulus scope is not established

Pages 6 and 36–37 · Theorem 8 and Section 5.4 · arXiv:2603.14116v2

The proof applies Corollary 16 and Lemma 20 to composite moduli. Their derivation uses a main term that omits the density of units, and the corrected factor is not propagated through the displayed lower bound and parameter choices. This is confined to the affected composite scope: it is not a counterexample to Theorem 8 and does not by itself classify the prime branch as incorrect.

Corollary 1Correct

The Zaremba existence statement is independently verified

Page 2 · Corollary 1 and Equation (5) · arXiv:2603.14116v2

Xin Zhang's later Theorem 1.2 proves the stronger assertion for every denominator using a corrected composite-modulus input. Corollary 1 therefore passes the statement-correctness test through a verifiable independent proof, although the route printed in this manuscript is incomplete. Because Zhang's paper appeared after the novelty cutoff, it is used here only to verify truth and not as evidence of non-novelty.

Xin Zhang, Zaremba's conjecture for all denominators ↗

02Proofs8 reported findingsContains incorrect or incomplete proofs

The composite-modulus mixing formulas are incorrect as written and leave their downstream branches incomplete. Independently, Lemma 32 and Proposition 35 contain unresolved obligations affecting the constructions used for Theorems 6–7. A separate false intermediate estimate in Proposition 35 has a verified replacement. Four literal notation defects are reported in yellow.

Lemmas 14–15 and Corollary 16Incorrect as written

The composite-modulus mixing estimates have the wrong main term

Pages 9–10 · Equations (32)–(34) · arXiv:2603.14116v2

Take $q=2^{Lm}$ , $N=2^m$ with a fixed sufficiently large $L$ , and $A=B=\mathbb Z/q\mathbb Z$ . For each $c$ , the congruence $(a+2c)(b+2c)\equiv1\pmod q$ has exactly $\varphi(q)=q/2$ pairs, so Lemma 14's exact count is $Nq/2$ , whereas its main term is $Nq$ and its error is $O(qN^{1-\kappa})$ . The discrepancy dominates that error. With $g(x)=x^{-1}$ and $S=[N]^2$ , Lemma 15 has the same missing unit density: the discrepancy is $qN^2/2$ , while its error is $O(qN^{2-2\kappa})$ . For Corollary 16, let $N\mid q$ , $I=[N]$ , and $\Lambda=\{0,N,\ldots,q-N\}$ , so $I\mathbin{\dotplus}\Lambda=\mathbb Z/q\mathbb Z$ ; then the exact inverse intersection has size $\varphi(q)=q/2$ , not the printed main term $q$ . Immediate-consequence closure cannot change this density. Repair classification: No repair supplied; a corrected local density must be propagated through Lemmas 17–20 and the composite branches of the main proofs.

Xin Zhang, corrected composite-modulus formulation ↗

Lemma 32Incomplete as written

Equation (107) drops an uncontrolled term

Pages 25–27 · Equations (91), (107), and (109)–(110) · arXiv:2603.14116v2

Write $x=x_{s_1}$ and $X=x_{s_*+\widetilde s_*}$ . The two inequalities in (91), together with $\mathcal D_2\leq D$ and $|\gamma_X|\leq C_3$ , give only $|\gamma_X^*|\leq(C_3/|\gamma_{s_1}|)(y/x+DC_3x/X)$ . Equation (107) retains only a multiple of the second term. That requires an additional estimate comparable to $yX\leq3DC_3x^2$ , which is neither assumed nor proved. Equations (109)–(110) use the reduced bound decisively. Repair classification: No repair supplied; the omitted $y/x$ term must be controlled or the contradiction reorganized.

Proposition 35Incomplete as written

Lemma 32's required coefficient bounds are chosen after its output

Pages 29–30 · Invocation of Lemma 32 through Equations (123)–(128) · arXiv:2603.14116v2

Lemma 32 requires a fixed vector $\vec C=(C_1,C_2,C_3)$ before producing $a,b,c,x,y,z$ . Proposition 35 invokes the lemma first and then defines $C_1,C_2,C_3$ from $X_1,X_2,X_3$ , which themselves depend on the resulting denominators and points. A statement of the form “for every fixed $\vec C$ , there exists an output” does not supply an output for coefficient bounds chosen from that same output. Immediate-consequence closure cannot exchange these quantifiers. The proof also replaces the returned denominator $y_0$ by a later $y$ without proving that the required independence is preserved. Repair classification: No repair supplied; a uniform, discretized, or simultaneous-selection argument is required.

Proposition 35, Equation (121)Incorrect as written · verified repair

The printed criticality estimate is unavailable, but the needed bound is repairable

Page 29 · Equation (121) · arXiv:2603.14116v2

The displayed estimate $|ax|\leq q/(\widetilde Mx_{s_0})$ would require $x_{s_0}$ to be $\widetilde M$ -critical, whereas it is selected only as an ordinary critical denominator. The conclusion needed in (121) nevertheless has a verified replacement. A convergent denominator gives $|ax|\leq q/x$ . From (120), $|I_*|\geq\delta^{60}|I|/5$ , $x\geq\sqrt{Nq}/H$ , $H=N^{9/20}$ , and $\widetilde M\leq N^{1/1000}$ , one obtains $|I_*|x^2/q\geq4N^{1/5}/\widetilde M>1$ . Hence $q/x\leq|I_*|x$ and $|bx|\leq|ax|+|b-a|x\leq2|I_*|x$ . This repairs the required final bound but not the other independent gaps in Proposition 35.

Proposition 35 statementTypo

The conclusion names a denominator instead of a partial quotient

Page 29 · Sentence following Equation (120) · arXiv:2603.14116v2

The proposition prints that for some denominator $q_j(a)\in[\sqrt{qN}/H,\sqrt{qN}]$ one has $q_j(a)\geq M_*$ . Replace the latter assertion by: the next partial quotient associated with such a denominator satisfies $c_{j+1}(a)>M_*$ . The proof immediately assumes the negation, namely that all partial quotients attached to denominators in this range are bounded by $M_*$ . This uniquely fixes the object named in the conclusion and changes no intended argument.

Lemma 32 notationTypo

The points attached to $y$ and $z$ are misidentified

Page 25 · Paragraph following Equation (103) · arXiv:2603.14116v2

The text prints $y=y(a),z=z(b)$ . Replace it by $y=y(b),z=z(c)$ . Lemma 32's statement and the immediately preceding selection $a<b<c$ determine the correction uniquely.

Proposition 35 notationTypo

The convergent has the wrong argument

Page 30 · Paragraph following Equation (128) · arXiv:2603.14116v2

The text prints $x=q_s(s)$ . Replace it by $x=q_s(a)$ . The subscript $s$ is the convergent index and $a$ is the rational whose convergent is used, so the correction is mechanical.

Theorem 7, Equation (19)Typo

A stray multiplication symbol should be deleted

Page 5 · Equation (19) · arXiv:2603.14116v2

The lower bound is printed as $q^{2w_M-1-o(1)}\cdot$ with no following factor. Delete the trailing $\cdot$ . The intended expression is $q^{2w_M-1-o(1)}$ , and no later calculation contains an additional factor at this point.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper: arXiv:2603.14116v2
Authors listed: Ilya D. Shkredov
Audit date: August 15, 2026

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P(delta)

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On some results of Korobov and Larcher and Zaremba's conjecture

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

The full stated conclusions are not established by the printed proof

The stated composite-modulus scope is not established

The Zaremba existence statement is independently verified

The composite-modulus mixing estimates have the wrong main term

Equation (107) drops an uncontrolled term

Lemma 32's required coefficient bounds are chosen after its output

The printed criticality estimate is unavailable, but the needed bound is repairable

The conclusion names a denominator instead of a partial quotient

The points attached to yyy and zzz are misidentified

The convergent has the wrong argument

A stray multiplication symbol should be deleted

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The points attached to $y$ and $z$ are misidentified