Discrete Fourier Restriction via Efficient Congruencing: Basic Principles

We show that whenever s > k(k + 1), then for any complex sequence (an)n∈Z, one has ∫ [0,1)k ∣∣∣∣ ∑ |n|6N ane(α1n+ . . .+ αkn ) ∣∣∣∣2s dα Ns−k(k+1)/2( ∑ |n|6N |an| )s . Bounds for the constant in the associated periodic Strichartz inequality from L to l of the conjectured order of magnitude follow, and likewise for the constant in the discrete Fourier restriction problem from l to L ′ , 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.