Additive equations in dense variables via truncated restriction 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 $(\log N{)}^{-c(\mathbf{P},\lambda )}$ 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 $\mathbf{P}=(x_1,\dots ,x_d,x_1^2+\cdots +x_d^2)$ that rely on the recent Strichartz estimates of Bourgain and Demeter.

41 pages

References in corpus (4)

• The cubic case of the main conjecture in Vinogradov's mean value theorem
• Proof of the main conjecture in Vinogradov's mean value theorem for degrees higher than three
• Discrete Fourier restriction via efficient congruencing: basic principles
• Some refinements for translation invariant quadratic forms in dense sets

Cited by in corpus (2)

• Discrete Fourier restriction via efficient congruencing: basic principles
• Decoupling for moment manifolds associated to Arkhipov--Chubarikov--Karatsuba systems