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A 1/64 spectral gap for surfaces with δ=1/2
Abstract
We show that any convex co-compact hyperbolic surface with exponent of convergence of Poincaré series δ∈ (2/5,14/27) has an essential spectral gap of size β=7/8(1/2−δ)+1/32δ−ε for any ε>0. In particular, for δ=1/2 this becomes β=1/64−ε. We show existence of the gap by proving a new Fractal Uncertainty Principle, using a two-ends Furstenberg theorem of O'Regan-Wu-Yi [arXiv:2607.08461].
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BibTeX
@misc{cohen2026spectral,
title = {{A 1/64 spectral gap for surfaces with $δ=1/2$}},
author = {Alex Cohen and Semyon Dyatlov},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.30261},
eprint = {2609.30261},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.30261},
}
RIS
TY - UNPB
TI - A 1/64 spectral gap for surfaces with δ=1/2
AU - Alex Cohen
AU - Semyon Dyatlov
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2609.30261
UR - https://arxiv.org/abs/2609.30261
AB - We show that any convex co-compact hyperbolic surface with exponent of convergence of Poincaré series δ∈ (2/5,14/27) has an essential spectral gap of size β=7/8(1/2−δ)+1/32δ−ε for any ε>0. In particular, for δ=1/2 this becomes β=1/64−ε. We show existence of the gap by proving a new Fractal Uncertainty Principle, using a two-ends Furstenberg theorem of O'Regan-Wu-Yi [arXiv:2607.08461].
ER -