Additive equations in dense variables via truncated restriction estimates

TL;DR

This paper investigates the solvability of translation-invariant additive equations in dense subsets of integer grids, using truncated restriction estimates and energy increment methods, extending results for polynomial and quadratic systems.

Contribution

It introduces new methods to analyze additive equations in dense sets, applying weak restriction estimates and recent Strichartz bounds to obtain positive results.

Findings

Positive results for polynomial systems in dense subsets

Extension to multidimensional quadratic systems

Use of weak restriction and Strichartz estimates

Abstract

We study translation-invariant additive equations of the form $\sum_{i=1}^s \lambda_i \mathbf{P}(\mathbf{n}_i) = 0$ in variables $\mathbf{n}_i \in \mathbb{Z}^d$, where the $\lambda_i$ are nonzero integers summing to zero, and $\mathbf{P}$ is a system of homogeneous polynomials such that the above equation is invariant by translation. We investigate the solvability of this equation in subsets of density $(\log N)^{-c(\mathbf{P},\mathbf{\lambda})}$ of a large box $[N]^d$, via the energy increment method. We obtain positive results in roughly the number of variables currently needed to derive a count of the solutions in the complete box $[N]^d$, for the curve $\mathbf{P} = (x,\dots,x^k)$ and the multidimensional systems of large degree studied by Parsell, Prendiville and Wooley, using only a weak form of restriction estimates. We also obtain results for the $(d+1)$-dimensional parabola $\mathbf{P}=(x_1,\dots,x_d,x_1^2+\dotsb+x_d^2)$ that rely on the recent Strichartz estimates of Bourgain and Demeter.