Discrete Fourier restriction via efficient congruencing: basic principles

Abstract

We show that whenever $s>k(k+1)$, then for any complex sequence $(\mathfrak a_n)_{n\in \mathbb Z}$, one has $$\int_{[0,1)^k}\left| \sum_{|n|\le N}\mathfrak a_ne(\alpha_1n+\ldots +\alpha_kn^k) \right|^{2s}\,{\rm d}{\mathbf \alpha}\ll N^{s-k(k+1)/2}\biggl( \sum_{|n|\le N}|\mathfrak a_n|^2\biggr)^s.$$ Bounds for the constant in the associated periodic Strichartz inequality from $L^{2s}$ to $l^2$ of the conjectured order of magnitude follow, and likewise for the constant in the discrete Fourier restriction problem from $l^2$ to $L^{s'}$, where $s'=2s/(2s-1)$. These bounds are obtained by generalising the efficient congruencing method from Vinogradov's mean value theorem to the present setting, introducing tools of wider application into the subject.