Additive equations in dense variables via truncated restriction estimates

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.