On Vus thin basis theorem in Warings problem

V. Vu has recently shown that when (kappa)(more than)2 and s is sufficiently large in terms of (kappa), then there exists a set X ((kappa)), 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 (kappa)th powers of elements of X ((kappa)) in order log n ways. We establish this conclusion with s~ (kappa) log (kappa), improving on the constraint implicit in Vuʹs wor(kappa) which forces s to be as large as (kappa)^48^(kappa). Indeed, the methods of this paper show, roughly spea(kappa)ing, that whenever existing methods permit one to show that all large integers are the sum of H((kappa)) (kappa)th powers of natural numbers, then H((kappa))+2 variables suffice to obtain a corresponding conclusion for "thin sets," in the sense of Vu.