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Additive equations in dense variables via truncated restriction estimates (1508.05923v5)

Published 24 Aug 2015 in math.CO and math.NT

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 $\mathbf{P} = (x,\dots,xk)$ 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 $\mathbf{P}=(x_1,\dots,x_d,x_12+\dotsb+x_d2)$ that rely on the recent Strichartz estimates of Bourgain and Demeter.

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