We show that whenever s>k(k+1), then for any complex sequence (anβ)nβZβ, one has β«[0,1)kβββ£nβ£β€Nββanβe(Ξ±1βn+β¦+Ξ±kβnk)β2sdΞ±βͺNsβk(k+1)/2(β£nβ£β€Nβββ£anββ£2)s. Bounds for
the constant in the associated periodic Strichartz inequality from L2s to
l2 of the conjectured order of magnitude follow, and likewise for the
constant in the discrete Fourier restriction problem from l2 to Lsβ²,
where sβ²=2s/(2sβ1). 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.Comment: 37 page