Asymptotic statistics for finite continued fractions with restricted digits

1. Problem Statement and Context

The paper addresses Zaremba's Conjecture, a central problem in Diophantine approximation. The conjecture posits that for any integer $N$, there exists a numerator $a$ coprime to $N$ such that the continued fraction expansion of the rational number $a/N$ has all partial quotients (digits) bounded by an absolute constant $A$.

• The Core Challenge: While the conjecture is elementary in statement, proving it for specific $N$ is difficult. Previous progress (e.g., Bourgain–Kontorovich) established the conjecture for $A=50$ for "almost every" $N$ using heavy analytic machinery (circle method, thin group orbits).
• The Reformulation: The author reframes the problem using the Hausdorff dimension ($\delta_A$) of the fractal set $E_A$ of real numbers in $[0,1]$ whose continued fraction digits are all $\le A$.
  • Let $\Sigma_{N,A}$ be the set of rationals $a/N$ with bounded digits.
  • Let $\Omega_{N,A}$ be the set of rationals $a/n$ with bounded digits where $n \le N$.
  • Conjecture 1.3 suggests that $|\Sigma_{N,A}| \sim N^{2\delta_A - 1}$ (for prime $N$).
• The Gap: Existing bounds (e.g., Moshchevitin–Murphy–Shkredov) provide estimates with exponents slightly larger than the conjectured $2\delta_A - 1$. The paper aims to refine these asymptotic estimates using dynamical systems methods.

2. Methodology

The author employs a dynamical approach based on Transfer Operators and Spectral Theory, extending the framework of Baladi–Vallée and Lee–Sun. The methodology consists of four main pillars:

A. Constrained Transfer Operators

The study utilizes the Gauss map $T(x) = \{1/x\}$ (and its complex analogue). The author defines transfer operators $\mathcal{L}_{s,w,A}$ constrained to a finite set of inverse branches corresponding to digits $\le A$.

• Spectral Properties: The operators act on $C^1$ function spaces. Key properties include:
  • Quasi-compactness: Existence of a unique simple dominant eigenvalue $\lambda_{s,w,A}$.
  • Analyticity: The eigenvalue and eigenfunction depend analytically on parameters near the Hausdorff dimension $\delta_A$.
  • Dolgopyat Estimates: Uniform polynomial decay of the resolvent norm $\|(I - \mathcal{L})^{-1}\|$ for large imaginary parts of the spectral parameter. This is crucial for controlling error terms.

B. Complex Function Analysis (Tauberian Theorems)

The author constructs a Dirichlet series $L(s,w) = \sum d_n(w) n^{-s}$, where $d_n(w)$ counts rationals with specific properties.

• Using the Implicit Function Theorem, a pole $s_0(w)$ is identified where the dominant eigenvalue $\lambda_{s_0(w), w, A} = 1$.
• Perron's Formula: A truncated Perron formula (with error estimates) is applied to invert the Dirichlet series. This converts the analytic properties of $L(s,w)$ (specifically the location of the pole $s_0(w)$) into asymptotic statistics for the counting function $\sum_{n \le N} d_n(w)$.

C. Probabilistic Smoothing Process

A critical technical hurdle is that the asymptotic formula derived via Perron's formula applies to the cumulative sum $\Omega_{N,A}$ (all $n \le N$), not the exact set $\Sigma_{N,A}$ (where $n=N$).

• To bridge this gap, the author introduces an auxiliary probability space $\Sigma_{N,A}(\epsilon)$, which is an "$\epsilon$-thickening" of the set $\Sigma_{N,A}$.
• This set includes rationals with denominators in the range $[N - \lfloor N\epsilon \rfloor, N]$.
• By analyzing the difference $\Psi_w(N) - \Psi_w(N - \lfloor N\epsilon \rfloor)$, the author derives the asymptotic behavior for this "thickened" set.

D. Generalization to Complex Continued Fractions

The methodology is extended to imaginary quadratic fields $\mathbb{Q}(\sqrt{-d})$ for $d \in \{1, 2, 3, 7, 11\}$.

• The Gauss map is generalized to the complex domain using Hurwitz's algorithm.
• The transfer operators are defined on a piecewise $C^1$ space over a Markov partition to handle the non-trivial admissibility conditions of complex digits.
• The spectral gap and Dolgopyat estimates are adapted to this higher-dimensional setting.

3. Key Contributions and Results

Main Theorem for Real Continued Fractions (Theorem 1.4)

The author proves the existence of an auxiliary set $\Sigma_{N,A}(\epsilon)$ such that:
$$\Sigma_{N,A} \subset \Sigma_{N,A}(\epsilon) \subset \Omega_{N,A}$$
and the size of this set satisfies the asymptotic estimate:
$$|\Sigma_{N,A}(\epsilon)| \sim N^{2\delta_A - \eta}$$
for some $0 < \eta < 1$.

• Significance: This reduces the exponent in the asymptotic formula from the trivial bound towards the conjectured $2\delta_A - 1$. It provides strong evidence for Zaremba's conjecture in an averaging sense.

Main Theorem for Complex Continued Fractions (Theorem 1.6)

Analogous results are established for complex continued fractions over imaginary quadratic fields:
$$|\Sigma_{N,A}(\epsilon)| \sim N^{2\delta_A - \theta}$$
for some $0 < \theta < 2$.

• Note: The exponent range is slightly weaker ($0 < \theta < 2$) compared to the real case ($0 < \eta < 1$) because the Hausdorff dimension $\delta_A$ in the complex case ranges from 0 to 2. The author suggests $\theta$ could potentially be improved to $1 < \theta < 2$ with a more subtle choice of $\epsilon(N)$.

Technical Refinements

• Smoothing: The paper successfully isolates the contribution of the "thickened" denominator range, showing that the asymptotic behavior is stable under small perturbations of the denominator.
• Spectral Gap Application: The work demonstrates the power of spectral gap estimates (Dolgopyat) in number-theoretic counting problems, moving beyond simple spectral radius arguments.

4. Significance and Implications

1. Averaging Sense Validation: While the paper does not prove Zaremba's conjecture for every $N$, it provides rigorous asymptotic evidence that the set of "good" rationals is large enough to satisfy the conjecture on average. It confirms that the density of such numbers follows the predicted power law dictated by the fractal dimension.
2. Unification of Real and Complex Cases: The paper successfully generalizes the dynamical approach from real continued fractions to complex continued fractions over imaginary quadratic fields, a non-trivial extension due to the lack of a simple ordering and more complex admissibility rules.
3. Methodological Advancement: It refines the application of Perron's formula combined with transfer operator spectral analysis. By introducing the smoothing process, the author overcomes the difficulty of extracting pointwise statistics from cumulative spectral data.
4. Connection to Fractal Geometry: The results reinforce the deep link between the arithmetic properties of rational numbers (Zaremba's conjecture) and the geometric properties of the underlying fractal sets (Hausdorff dimension).

In summary, Jungwon Lee provides a sophisticated dynamical proof that the number of rationals with bounded continued fraction digits behaves asymptotically as predicted by the Hausdorff dimension of the corresponding fractal set, both in the real and complex domains, effectively supporting Zaremba's conjecture in an averaged statistical framework.