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 $${\int }_{[0,1{)}^k}{\left|\sum _{|n|\le N}{\mathfrak{a}}_{n}e({\alpha }_{1}n+\dots +{\alpha }_k{n}^k)\right|}^{2s}\mathrm{d}\alpha \ll N^{s-k(k+1)/2}(\sum _{|n|\le N}|{\mathfrak{a}}_n{|}^2{)}^s.$$ 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 $L^{s^{\prime }}$, where $s^{\prime }=2s/(2s-1)$. 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.

37 pages

References in corpus (3)

• The cubic case of the main conjecture in Vinogradov's mean value theorem
• Additive equations in dense variables via truncated restriction estimates
• Some refinements for translation invariant quadratic forms in dense sets

Cited by in corpus (7)

• Nested efficient congruencing and relatives of Vinogradov's mean value theorem
• Decoupling for moment manifolds associated to Arkhipov--Chubarikov--Karatsuba systems
• Additive equations in dense variables via truncated restriction estimates
• Discrete restriction for $(x,x^3)$ and related topics
• Restricted mean value theorems and metric theory of restricted Weyl sums
• Discrete multilinear spherical averages
• Translation invariant quadratic forms and dense sets of primes