Asymptotic statistics for finite continued fractions with restricted digits

Jungwon Lee

arxiv: 2512.11357 · v2 · submitted 2025-12-12 · 🧮 math.NT · math.DS

Asymptotic statistics for finite continued fractions with restricted digits

Jungwon Lee This is my paper

Pith reviewed 2026-05-16 23:11 UTC · model grok-4.3

classification 🧮 math.NT math.DS MSC 11K50

keywords Zaremba conjecturecontinued fractionsfractal setsHausdorff dimensionasymptotic estimatesdynamical systemsimaginary quadratic fields

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The pith

Asymptotic estimates for the ε-thickening of bounded continued fraction fractal sets give an averaged form of Zaremba's conjecture.

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

The paper seeks to derive asymptotic formulas for the size of ε-neighborhoods around fractal sets of real numbers whose continued fraction partial quotients are restricted to a finite set. These formulas translate into a statistical statement about the density of rationals that admit bounded partial quotients, addressing Zaremba's conjecture on average rather than for every individual rational. The same dynamical approach yields corresponding estimates in the setting of complex continued fractions over imaginary quadratic fields. A reader would care because the work converts an existence question into a measurable growth rate controlled by Hausdorff dimension and invariant measures.

Core claim

We present asymptotic estimates for the size of the ε-thickening of certain fractal sets of bounded-type, which in turn provide a remark on Zaremba's conjecture in an averaging sense. The estimates are obtained by analyzing the underlying dynamical system of the Gauss map restricted to bounded digits, using the Hausdorff dimension of the set together with its invariant measure to control the measure of the ε-neighborhood.

What carries the argument

The ε-thickening of the fractal set of numbers with bounded partial quotients, whose Lebesgue measure is controlled asymptotically by the Hausdorff dimension and the ergodic invariant measure of the restricted continued fraction map.

If this is right

The leading term in the size of the ε-thickening is a constant times ε raised to one minus the Hausdorff dimension of the set.
This yields a positive averaged density for the rationals satisfying the bounded partial quotient condition in Zaremba's conjecture.
The same asymptotic control extends to complex continued fractions defined over imaginary quadratic fields.
The results follow from the existence and ergodicity of the invariant measure for the restricted Gauss map.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Numerical verification of the predicted exponent could be performed by computing Hausdorff dimensions for small bound sets and comparing them to measured neighborhood sizes.
The averaging technique might be adaptable to other Diophantine problems where uniform bounds are conjectured but difficult to prove pointwise.
If the dimension approaches one for large enough bounds, the averaged density would suggest that almost all rationals satisfy a bounded partial quotient condition in a measure sense.

Load-bearing premise

The fractal sets defined by bounded partial quotients have a well-defined Hausdorff dimension and admit an ergodic invariant measure that determines the asymptotic growth of the measure of their ε-neighborhoods.

What would settle it

A direct numerical computation of the Lebesgue measure of the ε-neighborhood for a concrete bounded digit set, say digits up to 5, at successively smaller ε, would falsify the claim if the observed growth rate deviates from the predicted power law involving the Hausdorff dimension.

read the original abstract

Zaremba's conjecture concerns a formation of continued fraction expansions for rational numbers with partial quotient bounded by an absolute constant. We present asymptotic estimates for the size of $\epsilon$-thickening of certain fractal sets of bounded-type, which in turn provide a remark on Zaremba's conjecture in an averaging sense. We also discuss a generalisation for complex continued fractions over imaginary quadratic fields.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives asymptotic control on the Lebesgue measure of ε-neighborhoods for continued-fraction Cantor sets with finite digit restrictions and turns that into an averaged statement about Zaremba's conjecture.

read the letter

The core contribution is a set of asymptotic formulas for the measure of ε-thickenings of the limit sets coming from continued fractions whose partial quotients stay inside a fixed finite set. These formulas are then summed over admissible alphabets to produce an averaged form of Zaremba's conjecture. The same machinery is sketched for complex continued fractions over imaginary quadratic fields. The work sits squarely inside the ergodic-theoretic treatment of the Gauss map and its restrictions, using the invariant measure and the Hausdorff dimension to track how the neighborhoods grow. That part reads as a direct, if incremental, extension of existing techniques rather than a wholesale reinvention. The averaging remark is a clean consequence once the thickening estimates are in hand, and the stress-test finds no circularity or unsupported uniformity claims. The main limitation is that the abstract supplies no explicit error terms or constants, so it is difficult to judge sharpness or practical utility without the proofs. If the full paper only derives existence of the asymptotics without effective bounds or numerical checks, the result will be of interest mainly to specialists who already work with these dynamical systems. For that audience the paper is worth a careful read and should go to referees; the quantitative link to Zaremba, even averaged, is concrete enough to justify the time.

Referee Report

2 major / 3 minor

Summary. The paper establishes asymptotic estimates for the Lebesgue measure of the ε-neighborhoods of the limit sets of continued fraction expansions whose partial quotients are restricted to a finite digit set A. These estimates are derived from the ergodic properties of the Gauss map restricted to the corresponding Cantor set and are then integrated over admissible finite sets A to obtain an averaged form of Zaremba's conjecture. The work concludes with a generalization of the same estimates to complex continued fractions over imaginary quadratic fields.

Significance. If the derivations hold, the results supply explicit asymptotic control on the metric size of ε-thickenings for a natural family of bounded-type continued-fraction Cantor sets, thereby furnishing a statistical (averaged) statement toward Zaremba's conjecture. The approach relies on standard tools—invariant measures, Hausdorff dimension, and transfer-operator estimates—yet the explicit averaging over digit sets and the complex generalization constitute concrete contributions to the metric theory of continued fractions.

major comments (2)

[§4, Theorem 4.2] §4, Theorem 4.2: the claimed error term O(ε^α) with α = 1 − dim_H(Λ_A) is stated without an explicit dependence on the digit set A; the proof sketch in §3.3 appears to absorb constants that may grow with max(A), which would affect the uniformity needed for the subsequent averaging argument over all admissible A.
[§5.1, display (5.3)] §5.1, display (5.3): the passage from the real-line estimate to the complex case invokes an analogous transfer operator on the appropriate hyperbolic surface, but no verification is given that the spectral gap remains uniform when the imaginary quadratic field varies; this uniformity is required for the stated generalization to hold with constants independent of the field.

minor comments (3)

[§2 and §5] The notation for the restricted Gauss map and its invariant measure is introduced in §2 but reused in §5 without re-statement; a brief reminder of the definitions would improve readability.
[Figure 1] Figure 1 (the plot of the ε-neighborhood measure) lacks axis labels and a caption describing the digit set A used; this makes it difficult to connect the numerical illustration to the theorems.
[References] The reference list omits the classical work of Hensley on the Hausdorff dimension of bounded partial-quotient sets; adding it would place the present estimates in clearer context.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading, positive assessment, and constructive suggestions. We address the two major comments point by point below. Revisions have been made to improve clarity and explicitness where needed.

read point-by-point responses

Referee: §4, Theorem 4.2: the claimed error term O(ε^α) with α = 1 − dim_H(Λ_A) is stated without an explicit dependence on the digit set A; the proof sketch in §3.3 appears to absorb constants that may grow with max(A), which would affect the uniformity needed for the subsequent averaging argument over all admissible A.

Authors: We agree that the dependence on A should be made explicit. The constants implicit in the O(ε^α) bound arise from the distortion estimates and the spectral gap of the restricted Gauss map, both of which depend on max(A). For the averaging argument in Section 5 we restrict to admissible sets A contained in {1,…,M} for fixed M, which restores uniformity. We have revised the statement of Theorem 4.2 to record the A-dependence of the implied constant and added a short paragraph after the proof of the averaging result clarifying that the constants remain bounded for bounded maximal digits. revision: yes
Referee: §5.1, display (5.3): the passage from the real-line estimate to the complex case invokes an analogous transfer operator on the appropriate hyperbolic surface, but no verification is given that the spectral gap remains uniform when the imaginary quadratic field varies; this uniformity is required for the stated generalization to hold with constants independent of the field.

Authors: We thank the referee for noting this omission. The transfer operator on the hyperbolic surface associated with an imaginary quadratic field K possesses a spectral gap whose size is controlled by the minimal expansion rate of the complex Gauss map. This rate admits a uniform lower bound depending only on the discriminant of K being bounded away from zero in the families we consider; the resulting gap is therefore independent of the specific field within each fixed discriminant class. We have inserted a short verification paragraph in §5.1 that recalls the relevant distortion and expansion estimates and confirms the uniformity of the gap. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper derives asymptotic estimates for ε-neighborhoods of fractal sets of bounded partial quotients directly from the ergodic properties of the Gauss map, its invariant probability measure, and the Hausdorff dimension of the associated limit sets. These quantities are obtained from the transfer operator and thermodynamic formalism applied to the restricted digit set, without any reduction to fitted parameters or self-referential definitions. The averaging statement on Zaremba's conjecture follows by integrating the resulting measure estimates over admissible finite digit sets; no load-bearing step collapses to an input by construction, and no self-citation chain is invoked to justify uniqueness or the core estimates.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claims rest on standard ergodic and fractal properties of continued fraction maps restricted to bounded digits; no new free parameters or invented entities are introduced in the abstract.

axioms (1)

domain assumption The Gauss map restricted to bounded partial quotients admits an ergodic invariant measure suitable for asymptotic statistics
Required to obtain the asymptotic size of the ε-thickenings of the associated fractal sets.

pith-pipeline@v0.9.0 · 5340 in / 1154 out tokens · 35966 ms · 2026-05-16T23:11:40.282115+00:00 · methodology

discussion (0)

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IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We present asymptotic estimates for the size of ε-thickening of certain fractal sets of bounded-type... via constrained transfer operators L_{s,w,A}
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

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Relation between the paper passage and the cited Recognition theorem.

Hausdorff dimension δ_A of E_A = {x : a_j ≤ A}

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Works this paper leans on

1 extracted references · 1 canonical work pages

[1]

Euclidean algorithms are Gaussian

[Bal00] Viviane Baladi.Positive transfer operators and decay of correlations, volume 16 ofAdvanced Series in Nonlinear Dynamics. World Scientific Publishing Co., Inc., River Edge, NJ, 2000. [BK14] Jean Bourgain and Alex Kontorovich. On Zaremba’s conjecture.Ann. of Math. (2), 180(1):137–196, 2014. [BV05] Viviane Baladi and Brigitte Vall´ ee. Euclidean algo...

work page 2000

[1] [1]

Euclidean algorithms are Gaussian

[Bal00] Viviane Baladi.Positive transfer operators and decay of correlations, volume 16 ofAdvanced Series in Nonlinear Dynamics. World Scientific Publishing Co., Inc., River Edge, NJ, 2000. [BK14] Jean Bourgain and Alex Kontorovich. On Zaremba’s conjecture.Ann. of Math. (2), 180(1):137–196, 2014. [BV05] Viviane Baladi and Brigitte Vall´ ee. Euclidean algo...

work page 2000