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

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Pith's one-line read Zaremba's conjecture holds for every large prime denominator: some a/q has all partial quotients bounded by an absolute constant.

desk verdict Proves absolute-constant Zaremba for every large prime (and positive-density composites) via new critical-denominator independence; M huge but the argument holds. read the letter →

arxiv 2603.14116 v2 pith:7TNAM2ZK submitted 2026-03-14 math.NT math.CAmath.CO

Ilya D. Shkredov This is my paper · ORCID
classification math.NTmath.CAmath.CO MSC 11J7011K3811A5511B30
keywords ZarembaconjecturecontinuedfractionspartialquotientsKorobovboundLarchersumAhlfors-Davidsetsdiscrepancymodularexpansion
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Zaremba's conjecture asks whether every positive integer q admits a numerator a coprime to q whose continued-fraction expansion has all partial quotients bounded by a fixed absolute constant. The paper proves this for every sufficiently large prime q (and more generally for a positive-density set of denominators that includes all large primes, large square-free integers, and high prime powers). It also shows that the bound can be taken as small as O(sqrt(log q)) while still producing asymptotically the expected number of such numerators, and that the sum of the partial quotients can be kept as small as O(log q times sqrt(log log q)). These statements improve classical bounds of Korobov and Larcher and give concrete discrepancy estimates for lattice-point sequences used in numerical integration.

What carries the argument

Critical denominators of Type I (denominators of convergents lying near sqrt(q)) living inside Ahlfors-David intervals of the Cantor set of rationals with bounded partial quotients. Their near-independence (linear relations with small coefficients are forbidden) is combined with Diophantine repulsion to eliminate those denominators that would force a large partial quotient.

What would settle it

Compute, for a sequence of large primes p, the minimal possible max partial quotient M(a) over a coprime to p; if this minimal value tends to infinity, the absolute-constant claim is false. Alternatively, check whether the product theorem used for intervals of length p^{1/9} holds with a positive spectral gap.

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Extended reading notes

Core claim

For every q belonging to a positive-density set Z that contains all large primes, there exist absolute constants M >= 2 and an intermediate bound such that at least q to the power 2w_M - 1 - o(1) residues a coprime to q have every partial quotient of a/q bounded by M. The same circle of ideas yields the weaker but still absolute bound O(sqrt(log q)) with the expected count of numerators, and an analogous lower bound for numerators whose partial-quotient sum is O(log q * sqrt(log log q)).

Load-bearing premise

The argument treats quantitative expansion bounds for the modular group SL_2(Z/qZ) as black boxes; if the spectral gap fails for the short Ahlfors-David intervals that arise, the independence of critical denominators collapses.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

0 major / 4 minor

Summary. The paper proves several strengthenings of classical results on bounded partial quotients of rationals a/q. For q belonging to a positive-density set Z (large primes, large square-free integers, high prime powers), Theorem 8 / Corollary 1 asserts the existence of absolute constants M ≥ 2 and ℳ ≪ M < ℳ such that there are at least q^{2w_M-1-o(1)} coprime a with all partial quotients of a/q bounded by M. Theorem 7 gives the asymptotically expected count q^{2w_M-1-o(1)} of a with M(a) ≤ M+2 once M ≥ C √log q. Theorem 6 improves Larcher’s bound on the sum of partial quotients to O(log q · √log log q) and supplies a matching lower bound on the number of such a. The argument combines the Cantor/Ahlfors–David structure of Z_M(t) (Lemmas 12, 20), expansion estimates in SL_2(Z/qZ) (Lemmas 13–16), a new Diophantine independence theory for critical denominators (Section 4, Lemmas 24, 27, Proposition 32), and a final repulsion step that eliminates denominators near √q.

Significance. If correct, the paper settles Zaremba’s conjecture for every sufficiently large prime (and for a positive-density set of composite moduli) with an absolute though non-effective bound M. This is a substantial advance over Korobov’s O(log q) bound and the author’s earlier O(log q / log log q) result, and it improves Larcher’s estimate on the sum of partial quotients. The new independence machinery for critical denominators (Lemmas 24, 27) and the Ahlfors–David analysis appear to be of independent interest for Diophantine approximation and fractal geometry. The lower bounds on the number of good numerators match the expected order of magnitude in the regime M = Ω(√log q), which is a clean and sharp feature of the method.

minor comments (4)
  1. The absolute constant M produced by Theorem 8 is acknowledged to be large and non-effective because of the expansion constant κ and the 1/9-threshold in (99) and (129). A short remark quantifying the dependence of M on κ (or stating that no explicit numerical bound is claimed) would help the reader.
  2. Notation for the two constants M and ℳ in Theorem 8 is slightly overloaded with the running parameter M used throughout Sections 2–5; a typographic distinction (e.g., script M versus roman M) would improve readability.
  3. Lemma 19 (Möbius inversion for reduced fractions) is used crucially for composite q; a one-sentence reminder that the same argument is vacuous for prime q would clarify the logical structure for readers interested only in the prime case.
  4. A few typographical slips appear (e.g., “Furthemore”, “acordingly”, missing spaces around some ≪ symbols). They do not affect the mathematics but should be cleaned in the final version.

Circularity Check

1 steps flagged · score 1.0 of 10

No significant circularity: expansion tools are self-cited but independent of the target bound; new Diophantine independence arguments stand alone.

  1. self citation load bearing [Section 2, Lemmas 13–14 and Corollary 15; Section 5.1–5.4]
    "The crucial result from [40, Lemma 4] … The proof is an application of the Bourgain–Gamburd machine [4], based on Helfgott’s expansion result [16]. … In the proof of Theorem 8 we also follow the argument from [40]"

    The quantitative expansion constant κ that powers the intersection estimates |A igcap B^{-1}| is taken from the author’s earlier papers rather than re-proved. While the cited results rest on external theorems (Helfgott, Bourgain–Gamburd) and do not presuppose the absolute-M conclusion, the present paper’s ability to reach an absolute (albeit large) M is load-bearing on those self-citations; without them the repulsion argument of Proposition 32 cannot be closed. This is ordinary methodological dependence, not a definitional loop, and therefore contributes only a minor score increment.

full rationale

The paper's central claims (absolute M for Zaremba numerators on the positive-density set Z, and the improved sum-of-quotients bound) are obtained by combining (i) the author's prior expansion estimates for SL_2(Z/qZ) (Lemmas 13–14, drawn from [40],[50] and ultimately Helfgott/Bourgain–Gamburd) with (ii) entirely new structural results on Ahlfors–David intervals, critical denominators, and their C-vector independence (Lemmas 24, 27, Proposition 32). The expansion lemmas supply a uniform spectral gap κ > 0 that is independent of the continued-fraction target; they do not encode or assume the existence of an absolute M. The new Diophantine repulsion argument then upgrades the earlier logarithmic bounds of Korobov/Larcher/[40] without any self-definitional loop, fitted parameter, or uniqueness theorem imported from the author's own work. The only self-citations are ordinary reuse of previously established analytic machinery; none of them force the final bound by construction. Hence the derivation is self-contained against external benchmarks and scores at most 1.

Assumptions & free parameters 3 free parameters · 4 assumptions · 3 invented entities

The central claims rest on three external black-box expansion results (Helfgott product theorem, Bourgain–Gamburd machine, affine sieve), on the classical theory of continued fractions and Hausdorff dimension of F_M, and on a handful of absolute constants that are never computed. No free parameters are fitted to data; the only 'parameters' are the absolute constants forced by the expansion machinery and the technical thresholds (1/9, 1/100, …) chosen for convenience. The invented entities (critical denominators of Type I/II, N-good intervals, δ-Assumption) are purely definitional tools internal to the proof.

free parameters (3)
  • absolute expansion constant κ
    Appears in Lemmas 13–16; its positive value is taken from the Bourgain–Gamburd machine and is never evaluated. All final exponents and the size of M depend on κ.
  • technical threshold 1/9 in N ≤ q^{1/9} = 1/9
    Chosen so that the Diophantine conditions (75),(99) hold; the paper notes that any positive power less than 1/2 would suffice in principle, but the concrete 1/9 is an artifact of the estimates.
  • absolute constant C in M ≥ C √ log q
    Appears in Theorem 7; large enough C absorbs all implicit constants from the expansion and independence lemmas.
assumptions (4)
  • domain assumption Helfgott's product theorem / expansion in SL_2(F_p) and its extensions to Z/qZ for q in Z
    Invoked as the engine of Lemmas 13–14; without a uniform spectral gap the intersection estimates |A igcap B^{-1}| fail.
  • standard math Hausdorff dimension formula w_M = 1 - 6/(π^{2} M) + O((log M)/M^{2}) of Hensley
    Used throughout to convert length estimates on intervals into cardinality estimates for Z_M(t).
  • standard math Classical correspondence between partial quotients and solutions of ax ≡ y (mod q) (Lemma 9)
    The bridge that turns modular-inverse statements into bounds on continued-fraction digits.
  • domain assumption Positive density of the set Z of admissible denominators
    Assumed so that the results apply to a positive-density set of q; the paper notes density at least 6/π^{2}.
invented entities (3)
  • critical denominators of Type I/II and M̃-critical denominators
    purpose: Isolate the intermediate convergents near √q whose linear independence yields the final repulsion bound.
    Definitional; no independent physical or arithmetic meaning outside the proof.
  • N-good intervals and δ-Assumption 21
    purpose: Guarantee that a positive proportion of subintervals of Z_M(t) still contain points with critical denominators after removing a negligible set.
    Technical bookkeeping device; independent evidence is not claimed.
  • Ahlfors–David structure of Z_M(t) independent evidence
    purpose: Provide uniform upper and lower density estimates on every scale that feed the independence lemmas.
    The structure is proved in Lemma 20 from classical continued-fraction geometry; it is not postulated but derived.

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Pith. "Pith review of On some results of Korobov and Larcher and Zaremba's conjecture." pith.science (2026). https://pith.science/paper/7TNAM2ZK

@misc{pith2026260314116,
  author       = {Pith},
  title        = {Pith review of: On some results of Korobov and Larcher and Zaremba's conjecture},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7TNAM2ZK}},
  note         = {Machine review of arXiv:2603.14116}
}
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 $\Omega(q^{1-O(1/\mathcal{M})})$ numbers $a$ coprime to $q$ such that all partial quotients of $a/q$ are bounded by $\mathcal{M}$.

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