Breakthrough work of Bourgain, Demeter, and Guth recently established that decoupling inequalities can prove powerful results on counting integral solutions to systems of Diophantine equations. In this note we demonstrate that in appropriate situations this implication can also be reversed. As a first example, we observe that a count for the number of integral solutions to a system of Diophantine equations implies a discrete decoupling inequality. Second, in our main result we prove an \(L^{2n}\) square function estimate (which implies a corresponding decoupling estimate) for the extension operator associated to a non-degenerate curve in \(\mathbb {R}^n\). The proof is via a combinatorial argument that builds on the idea that if \(\gamma \) is a non-degenerate curve in \(\mathbb {R}^n\), then as long as \(x_1,\ldots , x_{2n}\) are chosen from a sufficiently well-separated set, then \( \gamma (x_1)+\cdots +\gamma (x_n) = \gamma (x_{n+1}) + \cdots + \gamma (x_{2n}) \) essentially only admits solutions in which \(x_1,\ldots ,x_n\) is a permutation of \(x_{n+1},\ldots , x_{2n}\).

Bourgain，Demeter和Guth的突破性工作最近证明，解偶不等式可以证明对Diophantine方程组的积分解进行计数的强大结果。在本说明中，我们证明了在适当情况下这种含义也可以被颠倒。作为第一个示例，我们观察到对Diophantine方程组积分解的数量的计数意味着离散的解耦不等式。其次，在我们的主要结果中，我们证明了\（\ mathbb {R} ^ n中与非退化曲线相关联的扩展算子的\（L ^ {2n} \）平方函数估计（这意味着相应的解耦估计）\）。证明是通过组合论证建立的，该论证是如果\（\ gamma \）是\（\ mathbb {R} ^ n \）中的非简并曲线，则只要从充分分离的集合中选择\（x_1，\ ldots，x_ {2n} \），然后\（\ gamma（x_1）+ \ cdots + \ gamma（x_n）= \ gamma（x_ {n + 1}）+ \ cdots + \ gamma（x_ {2n}）\）本质上仅接受其中（（x_1，\ ldots） ，x_n \）是\（x_ {n + 1}，\ ldots，x_ {2n} \）的排列。