On Vu's thin basis theorem in Waring's problem

Abstract

V. Vu has recently shown that when k≥2 and s is sufficiently large in terms of k, then there exists a set $\mathfrak{X}$(k), whose number of elements up to t is smaller than a constant times (t log t)^1/s, for which all large integers n are represented as the sum of s kth powers of elements of $\mathfrak{X}$(k) in order log n ways. We establish this conclusion with s∼k log k, improving on the constraint implicit in Vu's work which forces s to be as large as k⁴8^k. Indeed, the methods of this paper show, roughly speaking, that whenever existing methods permit one to show that all large integers are the sum of H(k) kth powers of natural numbers, then H(k)+2 variables suffice to obtain a corresponding conclusion for "thin sets," in the sense of Vu.