We improve the range of ℓp(Zd)-boundedness of the integral k-spherical maximal functions introduced by Magyar. The previously best known bounds for the full k-spherical maximal function require the dimension d to grow at least cubicly with the degree k. Combining ideas from our prior work with recent advances in the theory of Weyl sums by Bourgain, Demeter, and Guth and by Wooley, we reduce this cubic bound to a quadratic one. As an application, we deduce improved bounds in the ergodic Waring--Goldbach problem.
|Number of pages||30|
|Publication status||Published - 29 May 2018|