小盘股
Small cap decouplings
摘要：
我们开发了一个工具箱，用于证明直径小于的规范规模的盒子。作为这种新技术的应用，我们解决了三个问题，早期的方法都失败了。我们从验证抛物线的小盘解耦开始。然后，我们发现尖锐的估计指数和小的频率分离的时刻曲线上的$\mathbb{R}^3$。这部分工作依赖于最近改进的Kakeya型平面管的估计，以及板和木板的新的多线性入射边界。我们还结合我们的方法与最近的进展，反向平方函数估计，以证明小帽解耦成平方样帽的二维锥。罗杰·希思-布朗的附录包含了矩曲线的新指数和估计对黎曼ζ函数的应用。
英文摘要：
We develop a toolbox for proving decouplings into boxes with diameter smaller than the canonical scale. As an application of this new technique, we solve three problems for which earlier methods have failed. We start by verifying the small cap decoupling for the parabola. Then we find sharp estimates for exponential sums with small frequency separation on the moment curve in $\mathbb{R}^3$. This part of the work relies on recent improved Kakeya-type estimates for planar tubes, as well as on new multilinear incidence bounds for plates and planks. We also combine our method with the recent advance on the reverse square function estimate, in order to prove small cap decoupling into square-like caps for the two dimensional cone. The Appendix by Roger Heath-Brown contains an application of the new exponential sum estimates for the moment curve, to the Riemann zeta-function.
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