[Paper Review] Fractal Uncertainty Principle with Explicit Exponent

Long Jin, Ruixiang Zhang|arXiv (Cornell University)|Sep 30, 2017

Mathematical Dynamics and Fractals11 references3 citations

TL;DR

This paper establishes an effective, explicit exponent β in the fractal uncertainty principle for Fourier transforms on Ahlfors-David regular sets, with dependence on the Hausdorff dimension δ and regularity constant C_R. The key result provides a double-exponential lower bound on the spectral gap for convex co-compact hyperbolic surfaces and open quantum baker’s maps, resolving an open problem on effective spectral gaps in quantum chaos.

ABSTRACT

We prove an explicit formula for the dependence of the exponent in the fractal uncertainty principle of Bourgain-Dyatlov on the dimension and on the regularity constant for the regular set. In particular, this implies an explicit essential spectral gap for convex co-compact hyperbolic surfaces when the Hausdorff dimension of the limit set is close to 1.

Motivation & Objective

• To provide an effective, quantitative dependence of the fractal uncertainty principle exponent β on the Hausdorff dimension δ and regularity constant C_R.
• To resolve the lack of explicit bounds in the original Bourgain–Dyatlov fractal uncertainty principle by making the exponent β computable.
• To establish an explicit essential spectral gap for convex co-compact hyperbolic surfaces when δ is close to 1.
• To extend effective bounds to open quantum baker’s maps and relate them to the geometry of the underlying Cantor set.
• To improve upon prior results by providing a constructive, non-contradiction-based proof using a refined version of the Beurling–Malliavin multiplier theorem.

Proposed method

• The authors use a variant of the Beurling–Malliavin multiplier theorem with a Lipschitz condition on the Hilbert transform of log ω, rather than on log ω itself, to achieve effective bounds.
• They derive an effective version of the multiplier theorem (Theorem 3.2) that controls the decay of the Fourier transform on fractal sets.
• The proof relies on a quantitative refinement of the Beurling–Malliavin theory, focusing on the Cartwright class of entire functions and Cauchy integrals.
• The method avoids contradiction-based arguments used in Bourgain–Dyatlov, enabling explicit dependence on δ and C_R.
• The construction is applied to Fourier integral operators and then to the resolvent of the Laplacian on hyperbolic surfaces.
• The spectral gap is derived via cutoff resolvent estimates and the relation between the fractal uncertainty principle and the absence of embedded eigenvalues.

Experimental results

Research questions

• RQ1 What is the explicit dependence of the fractal uncertainty principle exponent β on the Hausdorff dimension δ and regularity constant C_R for Ahlfors-David regular sets?
• RQ2 Can an effective, non-contradiction-based proof of the fractal uncertainty principle be constructed using refined multiplier theorems?
• RQ3 What is the size of the essential spectral gap for convex co-compact hyperbolic surfaces when δ is close to 1?
• RQ4 How does the spectral gap for open quantum baker’s maps depend on the base M and the alphabet size |A|?
• RQ5 Can the effective spectral gap be quantitatively compared to prior results, especially for special sequences N = M^k?

Key findings

• The paper establishes an explicit exponent β = exp[−exp(K(C_R δ^{-1}(1−δ)^{-1})^{K(1−δ)^{-2}})] for the fractal uncertainty principle, with K a universal constant.
• For convex co-compact hyperbolic surfaces, the essential spectral gap is bounded below by β_M = exp[−exp(K(C_R δ^{-1}(1−δ)^{-1})^{K(1−δ)^{-3}})], ensuring only finitely many resonances above −β_M.
• The cutoff resolvent estimates for the Laplacian on such surfaces hold uniformly in the spectral parameter λ with decay rate |λ|^{-1−2min(0,Imλ)+ε} for |Reλ| ≥ C₀.
• For open quantum baker’s maps with δ ≤ 1/2, the spectral gap satisfies β ≥ 1/2 − δ + (40M^{3δ})^{-160/(δ(1−δ))}, showing polynomial improvement over the pressure gap.
• For δ ≥ 1/2, the spectral gap is bounded below by β ≥ exp[−exp(KM^{K(1−δ)^{-2}})], a double-exponential improvement over the trivial gap β = 0.
• The results improve upon prior work by Dyatlov–Jin for special sequences N = M^k, where β ≥ 1/2 − δ + 1/(K M^8 log M) and β ≥ exp[−M^{δ/(1−δ)+o_M(1)}], showing stronger bounds for such sequences.