We give a new proof of $l^2$ decoupling for the parabola inspired from efficient congruencing. Making quantitative this proof matches a bound obtained by Bourgain for the discrete restriction problem for the parabola. We illustrate similarities and differences between this new proof and efficient congruencing and the proof of decoupling by Bourgain and Demeter. We also show where tools from decoupling such as $l^2L^2$ decoupling, Bernstein’s inequality, and ball inflation come into play.

中文翻译：

---

有效同余的 $l^2$ 解耦解释：抛物线

我们为受有效同余启发的抛物线提供了 $l^2$ 解耦的新证明。进行量化这个证明匹配 Bourgain 为抛物线的离散限制问题获得的界限。我们说明了这个新证明和有效同余与 Bourgain 和 Demeter 的解耦证明之间的异同。我们还展示了解耦工具（例如 $l^2L^2$ 解耦、伯恩斯坦不等式和球膨胀）的作用。