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.

我们开发了一个工具箱，用于将去耦器证明为直径小于标准刻度的箱子。作为这项新技术的应用，我们解决了早期方法失败的三个问题。我们首先验证抛物线的小电容去耦。然后，我们在\（\ mathbb {R} ^ 3 \）的矩曲线上找到具有较小频率间隔的指数和的精确估计。这部分工作依赖于最近对平面管的改进的Kakeya型估计以及板和木板的新的多线性入射边界。我们还将我们的方法与反向平方函数估计的最新进展相结合，以证明二维二维圆锥的小盖帽解耦为正方形盖帽。罗杰·希思·布朗（Roger Heath-Brown）的附录包含了矩曲线的新指数求和估计在黎曼zeta函数中的应用。