The Limiting Distribution of Elliptic Dedekind Sums

1. Problem Statement and Context

Classical Background:
The paper investigates the statistical distribution of Dedekind sums, defined for coprime integers $h, k$ as:
$$s(h, k) = \sum_{l=1}^{k-1} \left(\!\left( \frac{l}{k} \right)\!\right) \left(\!\left( \frac{hl}{k} \right)\!\right)$$
where $((x))$ is the sawtooth function. Classical results by Vardi established that, when suitably normalized, these sums follow a Cauchy distribution.

The Elliptic Generalization:
The authors focus on Sczech sums (or elliptic Dedekind sums), introduced by Sczech as generalizations to complex lattices associated with imaginary quadratic fields $K = \mathbb{Q}(\sqrt{-D})$. For coprime elements $a, c \in \mathcal{O}_K$ (the ring of integers), the sum is defined as:
$$D(a, c) = \frac{1}{c} \sum_{r \in L/cL} G_1\left(\frac{ar}{c}\right) G_1\left(\frac{r}{c}\right)$$
where $G_1$ is an elliptic analogue of the cotangent function. The normalized quantity $\tilde{D}(a, c)$ is rational and real.

The Core Question:
While the classical sums exhibit a Cauchy limit, the behavior of the elliptic sums was an open problem. Ito [30] conjectured that the average magnitude of these sums diverges, suggesting a different distributional behavior than the classical case. The paper aims to determine the limiting distribution of $\tilde{D}(a, c)$ as the norm of the denominator grows ($|c|^2 < X$) and to prove Ito's conjecture.

2. Methodology

The authors employ a sophisticated blend of dynamical systems, spectral theory, and analytic number theory. The approach diverges from the classical spectral methods used by Vardi (which relied on automorphic forms and Kloosterman sums) and instead utilizes a dynamical framework inspired by the work of Bettin, Drappeau, Baladi, and Vallée.

Key Methodological Steps:

1. Complex Continued Fractions & The Hurwitz Map:
   The authors utilize the continued fraction expansion of complex numbers in imaginary quadratic fields. They define the Hurwitz map $G(z) = \frac{1}{z} - [\frac{1}{z}]$ on fundamental domains $I_D$ (rectangular for $D=2$, hexagonal for $D=7, 11$). The Dedekind sum is expressed as a sum over the partial quotients (digits) of this expansion.

2. Transfer Operators and Spectral Analysis:
   The problem is translated into the study of a transfer operator (Ruelle-Perron-Frobenius operator) acting on a Banach space of holomorphic functions defined on a Markov partition of the fundamental domain.

   • They define a two-parameter family of operators $K_{s,t}$, where $s$ relates to the height of the fraction and $t$ is the Fourier variable for the characteristic function.
   • The characteristic function of the normalized sum is linked to a Dirichlet series $H(2s, t)$, which is expressed via the resolvent of the operator: $H(2s, t) \approx (I - K_{s,t})^{-1}$.

3. Meromorphic Continuation and Dolgopyat Estimates:
   To establish the limiting distribution, the authors must analyze the singularities of $H(2s, t)$.

   • They prove the meromorphic continuation of $H(2s, t)$ to a strip $\text{Re}(s) > 1 - \delta$.
   • Crucially, they establish a spectral gap and decay of correlations for large imaginary parts of $s$ (large $\tau$). This requires adapting Dolgopyat's estimates (originally for hyperbolic dynamics) to the number-theoretic setting of complex continued fractions. This step is necessary to control the error terms in the asymptotic expansion.

4. Asymptotic Expansion of the Characteristic Function:
   Using perturbation theory around the leading eigenvalue $\lambda(s, t)$, they derive an asymptotic expansion for the characteristic function $\chi_S(t)$ as $t \to 0$.

   • A critical technical hurdle is that the period function $\Psi(z)$ associated with the sum grows linearly with the digit size (unlike the logarithmic growth in classical Euclidean algorithms). This prevents the direct application of Hwang's quasi-power theorem.
   • Instead, they perform a detailed analysis of the oscillatory integral $\int (e^{it\Psi(z)} - 1) d\mu(z)$, where $\mu$ is the invariant measure of the dynamical system. This involves analyzing the geometry of level sets of the period function and the density of the invariant measure near the "cusps" of the fundamental domain.

3. Key Contributions and Results

Main Theorem (Theorem 1.3):
The paper proves that the normalized elliptic Dedekind sums follow a Gaussian (Normal) distribution, contrasting sharply with the Cauchy distribution of classical sums.

Specifically, for $D \in \{2, 7, 11\}$, let $K_{\bullet, D}(X)$ be the set of fractions $a/c$ with $1 \le |c|^2 < X$. There exists a constant $\kappa_D > 0$ such that for any $x \in \mathbb{R}$:
$$P_X \left( z \in K_{\bullet, D}(X) : \frac{\tilde{D}(z)}{\kappa_D \sqrt{\log X \log \log X}} < x \right) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^x e^{-y^2/2} dy + O\left( \frac{1}{(\log \log X)^{1-\epsilon}} \right)$$

Proof of Ito's Conjecture (Corollary 1.4):
As a direct consequence, the authors prove Ito's conjecture regarding the divergence of the average magnitude:
$$\lim_{X \to \infty} \frac{1}{\#(K_2(X) \cap B_2)} \sum_{a/c \in K_2(X) \cap B_2} |\tilde{D}(a, c)| = \infty$$
The growth rate is shown to be of order $\sqrt{\log X \log \log X}$.

Technical Innovations:

• Gaussian vs. Cauchy: The paper identifies the mechanism causing the shift from Cauchy to Gaussian. In the classical case, the heavy tails of the distribution arise from the specific structure of the sawtooth function and the spectral properties of the Gauss map. In the elliptic case, the specific geometry of the complex continued fraction expansion and the growth of the period function lead to a central limit theorem behavior.
• Handling Non-Moderate Growth: The authors successfully handle functions that do not satisfy the "moderate growth" condition ($f(x) \ll \log(1/x)$) required by previous theorems (like those of Baladi-Vallée), extending the dynamical methods to a broader class of arithmetic functions.
• Invariant Measure Analysis: They provide a precise asymptotic analysis of the invariant measure $\mu$ near the boundary of the fundamental domain, which is essential for evaluating the oscillatory integrals that determine the variance of the limiting distribution.

4. Significance

• Resolution of a Long-standing Conjecture: The paper settles Ito's conjecture regarding the unboundedness of the average elliptic Dedekind sums.
• New Distributional Paradigm: It establishes that elliptic Dedekind sums, despite being generalizations of the classical sums, belong to a different universality class (Gaussian vs. Cauchy). This highlights the profound impact of the underlying geometry (complex lattices vs. real line) on the statistical properties of arithmetic sums.
• Methodological Advancement: The work bridges the gap between the dynamical approach to Dedekind sums (Bettin-Drappeau) and the spectral theory of complex continued fractions (Kim-Lee-Lim). The successful application of Dolgopyat-type estimates to the specific geometry of imaginary quadratic fields provides a robust toolkit for future studies of modular symbols and L-functions in higher dimensions.
• Quantitative Precision: The paper provides explicit error terms and identifies the correct normalization factor ($\sqrt{\log X \log \log X}$), offering a complete quantitative description of the distribution.

In summary, this paper represents a major advancement in the analytic theory of Dedekind sums, demonstrating that the transition from real to complex lattices fundamentally alters the limiting distribution from Cauchy to Gaussian, and providing the rigorous dynamical machinery required to prove such results.