Additive equations in dense variables via truncated restriction estimates

作者：Henriot Kevin

来源：Proceedings of the London Mathematical Society, 2017, 114(5): 927-959.

DOI：10.1112/plms.12028

We study additive equations of the form Sigma(s)(i=1) lambda(i) P(n(i)) = 0 in variables n(i) is an element of Z(d), where the lambda(i) are nonzero integers summing up to zero and P = (P1,..., P-r) is a system of homogeneous polynomials making the equation translation-invariant. We investigate the solvability of this equation in subsets of density (logN)(-c(P,lambda)) of a large box [N] d via the energy-increment method. We obtain positive results for roughly the number of variables currently needed to derive a count of solutions in the complete box [N](d) for the multidimensional systems of large degree studied by Parsell, Prendiville and Wooley. Appealing to estimates from the decoupling theory of Bourgain, Demeter and Guth, we also treat the cases of the monomial curve P = (x,..., x(k)) and the parabola P = (x, vertical bar x vertical bar(2)) for a number of variables close to or equal to the limit of the circle method.

• 出版日期2017-4