Reversing a Philosophy: From Counting to Square Functions and Decoupling

Abstract

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 $x_{n+1},\ldots , x_{2n}$.