AI 中文总结
研究维度大于1/2的凸余紧肖特基群相关的帕特森 - 沙利文测度的傅里叶衰减，用基本的振荡积分估计等取代先前技术机制，得到明确衰减指数\(\frac{\delta(2\delta - 1)}{(2\delta + 1)(3 - \delta)}\)。
AI 中文摘要
我们给出了与维度\(\delta>\frac{1}{2}\)的凸余紧肖特基群相关的帕特森 - 沙利文测度的幂次傅里叶衰减的基本证明，得到了明确的衰减指数\(\frac{\delta(2\delta - 1)}{(2\delta + 1)(3 - \delta)}\)。证明用基本的振荡积分估计、双曲几何和基于转置肖特基群的对偶论证取代了先前工作中使用的技术机制，如和积估计、更新理论、多尔戈皮亚特定理和\(L^2\)平坦化。
英文摘要
We give an elementary proof of power Fourier decay for Patterson-Sullivan measures associated to convex co-compact Schottky groups of dimension $δ>\frac12$, obtaining the explicit decay exponent $\frac{δ(2δ- 1)}{(2δ+ 1)(3 - δ)}$. The proof replaces the technical machinery used in previous works, such as sum-product estimates, renewal theory, Dolgopyat methods, and $L^2$ flattening, by elementary oscillatory integral estimates, hyperbolic geometry, and a duality argument based on the transpose Schottky group.
Comments19 pages
