On Waring's Problem for Cubes and Smooth Weyl Sums

Abstract

Non-trivial estimates for fractional moments of smooth cubic Weyl sums are developed. Complemented by bounds for such sums of use on both the major and minor arcs in a Hardy-Littlewood dissection, these estimates are applied to derive an upper bound for the sth moment of the smooth cubic Weyl sum of the expected order of magnitude as soon as s> 7.691. Related arguments demonstrate that all large integers n are represented as the sum of eight cubes of natural numbers, all of whose prime divisors are at most exp (c(log nlog log n)1/2}, for a suitable positive number c. This conclusion improves a previous result of G. Harcos in which nine cubes are required. 1991 Mathematics Subject Classification: 11P05, 11L15, 11P55.