Discrete Fourier restriction via Efficient Congruencing

作者：Wooley Trevor D

来源：International Mathematics Research Notices, 2017, (5): 1342-1389.

DOI：10.1093/imrn/rnw031

We show that whenever s > k( k + 1), then for any complex sequence ( an) n. Z, one has
integral([0,1)k) vertical bar Sigma(vertical bar n vertical bar <= X) a(n)e(alpha(1)n + ... + alpha(k)n(k))vertical bar(2s) d alpha << Xs-k(k+1)/2(Sigma(vertical bar n vertical bar <= X) vertical bar a(n)vertical bar(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', 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.

• 出版日期2017-3