A 1/64 spectral gap for surfaces with $\delta =1\mathrm{/}2$

We show that any convex co-compact hyperbolic surface with exponent of convergence of Poincaré series $\delta \in (\frac{2}{5},\frac{14}{27})$ has an essential spectral gap of size $\beta =\frac{7}{8}(\frac{1}{2}-\delta )+\frac{1}{32}\delta -\epsilon$ for any $\epsilon >0$. In particular, for $\delta =\frac{1}{2}$ this becomes $\beta =\frac{1}{64}-\epsilon$. 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].