Discrete Fourier restriction via Efficient Congruencing

We show that whenever s > k(k + 1), then for any complex sequence (an)n∈Z, one has Z [0,1)k X |n|6X ane(α1n + . . . + αkn k ) 2s dα Xs−k(k+1)/2 X |n|6X |an| 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 0 , where s 0 = 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.