Quantitative Fourier decay for Patterson-Sullivan measures of dimension larger than $1/2$

Technical Summary: Quantitative Fourier Decay for Patterson-Sullivan Measures of Dimension Larger Than 1/2

Problem Statement
The paper addresses the problem of establishing quantitative Fourier decay for Patterson-Sullivan (PS) measures associated with convex co-compact Schottky groups $\Gamma \subset \text{PSL}_2(\mathbb{R})$. Let $\Lambda_\Gamma$ be the limit set of $\Gamma$, a Cantor set of Hausdorff dimension $\delta \in (0, 1)$. The PS measure $\mu$ is the unique $\delta$-conformal probability measure supported on $\Lambda_\Gamma$. The central question is whether the Fourier transform $\hat{\mu}(\xi) = \int e^{-2\pi i \xi x} d\mu(x)$ exhibits polynomial decay, i.e., $|\hat{\mu}(\xi)| \lesssim |\xi|^{-\eta}$ for some $\eta > 0$, and specifically, to obtain an explicit exponent $\eta$ as a function of $\delta$.

Previous breakthroughs by Bourgain and Dyatlov [13] established such decay for $\delta > 1/2$ using the sum-product theorem. Subsequent works generalized this to higher dimensions or provided alternative proofs using renewal theory, Dolgopyat methods, or $L^2$ flattening. However, these approaches rely on sophisticated machinery, and obtaining explicit, quantitative exponents has remained challenging.

Methodology
The authors present an elementary proof that avoids the technical machinery of sum-product estimates, renewal theory, and $L^2$ flattening. Instead, the proof combines:

1. Oscillatory Integral Estimates: Utilizing techniques introduced by Kaufman [18] and developed for Gibbs measures by Queffélec-Ramaré [36] and Jordan and Sahlsten [17].
2. Hyperbolic Geometry: Exploiting the bounded distortion properties of the Schottky group action and the structure of the limit set.
3. Duality Argument: A crucial innovation is the use of the "transpose Schottky group" $\Gamma^T$. The authors reformulate the non-concentration estimates required for the proof in terms of the PS measure of $\Gamma^T$. This relates to the "dual IFS" framework [11] and provides a clearer geometric interpretation of the non-linearity of the system.

The core of the argument involves approximating the Fourier transform by a sum of oscillatory integrals over cylinder sets defined by a stopping time $\tau$. The decay is derived by estimating the $L^2$ norm of these sums. The proof relies on a "non-concentration estimate" for the ratios of matrix coefficients ($c_a/d_a$) of the group elements, which is proven using the $\delta$-regularity of the PS measure of the transpose group.

Key Contributions and Results
The primary result is Theorem 1, which provides an explicit power-law decay exponent for $\delta > 1/2$:

$$|\hat{\mu}(\xi)| \leq C |\xi|^{-\eta}, \quad \text{where} \quad \eta = \frac{\delta(2\delta-1)}{(2\delta+1)(3-\delta)}.$$

The proof proceeds through the following steps:

• Reduction to Oscillatory Integrals: The Fourier transform is expressed as an integral of a function $f$ constructed from the group action. An approximation lemma (Lemma 3.1) reduces the problem to bounding the $L^2$ norm of $f$ and its derivative.
• $L^2$ Bound via Non-Concentration: The $L^2$ norm is expanded into a double sum over words $a, b \in W_\tau$. The phase function involves differences of group actions $\gamma_{a'}(x) - \gamma_{b'}(x)$. The authors apply integration by parts (Lemma 3.5) to bound these integrals, distinguishing cases based on the presence of critical points.
• Counting Small Differences: The bounds depend on how often the differences in matrix coefficients ($|c_{a'} - c_{b'}|$ or $|d_{a'} - d_{b'}|$) are small. Lemma 3.8 establishes a non-concentration bound for these differences, derived from the regularity of the transpose group's measure (Lemma 2.9).
• Optimization: By choosing the stopping time parameter $\tau$ and the approximation scale $\eta$ as functions of $|\xi|$, the authors optimize the resulting bounds to achieve the stated exponent.

Significance and Scope
The paper claims significance primarily for its elementary nature and the explicitness of the decay exponent.

• Elementary Proof: The authors demonstrate that for $\delta > 1/2$, the heavy machinery of sum-product theory and renewal theory is not strictly necessary. The proof relies on standard tools from harmonic analysis and hyperbolic geometry.
• Explicit Exponent: While the exponent obtained is not necessarily optimal (the authors note it differs slightly from Queffélec-Ramaré's exponent due to a sharper approximation lemma), it is fully explicit and derived without abstract existence arguments.
• Limitations: The method is restricted to $\delta > 1/2$. The authors note that the approximation procedure and oscillatory integral estimates at scale $|\xi|^{-1/2}$ do not appear to work below this threshold.
• Extensions: The authors suggest the method extends to geometrically finite groups with parabolic elements (where $\delta \geq 1/2$) and to Ahlfors-regular self-conformal measures satisfying a non-linearity condition on the dual IFS, provided the dimension exceeds a specific threshold.

Relation to Spectral Gap
The paper contextualizes the result within the Fractal Uncertainty Principle. It notes that while Fourier decay implies an improvement in the essential spectral gap of the Laplacian, the specific exponent derived here is not sufficient to prove the Jakobson-Naud conjecture (which predicts a gap of $1/2 - \delta + \delta/2$) even for $\delta$ near $1/2$. However, the authors state that improving their exponent would yield a better spectral gap bound.

In summary, the paper provides a streamlined, self-contained proof of quantitative Fourier decay for PS measures with $\delta > 1/2$, replacing complex combinatorial and dynamical systems tools with elementary oscillatory integral techniques and a duality argument involving the transpose group.