We prove an explicit formula for the dependence of the exponent $$\beta $$ β in the fractal uncertainty principle of Bourgain–Dyatlov (Ann Math 187:1–43, 2018) on the dimension $$\delta $$ δ and on the regularity constant $$C_R$$ C R 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.

中文翻译：

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具有显式指数的分形不确定性原理

我们证明了 Bourgain–Dyatlov 的分形不确定性原理（Ann Math 187:1–43, 2018）中指数 $$\beta $$ β 依赖于维度 $$\delta $$ δ 的显式公式。正则集的正则常数 $$C_R$$ CR。特别是，当极限集的 Hausdorff 维数接近 1 时，这意味着凸共紧双曲曲面的显式基本光谱间隙。