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

V. Vu has recently shown that when k greater than or equal to 2 and s is sufficiently large in terms of k, then there exists a set 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 X(k) in order log n ways. We establish this conclusion with s similar to k log k, improving on the constraint implicit in Vu's work which forces s to be as large as k(4)8(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.