Improved fractal Weyl bounds for convex cocompact hyperbolic surfaces and large resonance-free regions

2023/01/08 by Louis Soares, Soares, Louis · 1 citation
Mathematics · #37C30 (Secondary) #58J50 #81U24 (Primary) 11M36 #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #Spectral Theory (math.SP)

paper · pdf · doi:10.48550/arxiv.2301.03023

openalex publication_date 2023/01/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let X be a convex cocompact hyperbolic surface, and let δ denote the Hausdorff dimension of its limit set. Let N_X(σ,T) denote the number of resonances of X inside the box [σ, δ] + i[0,T]. We prove that for all σ> δ/2, we have N_X(σ,T) ≪_εT^{1 + δ- 2(2σ- δ) + ε}. This strengthens the previously established "improved" fractal Weyl bounds due to Naud \citeNaud14 and Dyatlov \citeDya19. Moreover, this result implies that for every ε> 0, there exist resonance-free rectangular boxes of arbitrary height within the strip \ s ∈ ℂ : \tfrac34δ+ εlt; Re(s) lt; δ \. Our proof combines Naud's approach \citeNaud14 with the refined transfer operator machinery developed by Dyatlov-Zworski \citeDyZw18, as well as a new estimate for oscillatory integrals that arise naturally in our analysis.

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