Fractal uncertainty for transfer operators

TL;DR

This paper demonstrates that the fractal uncertainty principle implies finiteness of zeros of the Selberg zeta function in a specific region for convex co-compact hyperbolic surfaces, simplifying previous microlocal approaches.

Contribution

It establishes a direct link between the fractal uncertainty principle and the distribution of zeros of the Selberg zeta function, bypassing complex microlocal techniques.

Findings

Finiteness of zeros with real part ≥ 1/2 - σ for some σ > 0

Simplification of proof techniques compared to previous microlocal methods

Potential implications for spectral theory of hyperbolic surfaces

Abstract

We show directly that the fractal uncertainty principle of Bourgain-Dyatlov [arXiv:1612.09040] implies that there exists $ \sigma > 0 $ for which the Selberg zeta function for a convex co-compact hyperbolic surface has only finitely many zeros with $ \Re s \geq \frac12 - \sigma$. That eliminates advanced microlocal techniques of Dyatlov-Zahl [arXiv:1504.06589] though we stress that these techniques are still needed for resolvent bounds and for possible generalizations to the case of non-constant curvature.