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Amongst our opera devoted to additive problems restricted to thin polynomial sequences (see in particular [3], [4], [5]), the tertiary part is devoted to estimates for exceptional sets associated with the expected asymptotic formula for the number of representations of prescribed type. While such estimates lead directly to lower bounds of the anticipated size for likewise restricted moments of the number of representations, uncertainties concerning integers associated with the exceptional set prohibit any immediate inference of asymptotic formulae for such moments. The purpose of this paper is to develop methods that establish such asymptotic formulae, thereby avoiding the aforementioned uncertainties. Continuing the tradition of our previous excursions in this series, we illustrate our ideas with a discussion of Waring’s problem for cubes. Denote by Rs(n) the number of representations of n as the sum of s cubes of positive integers. A heuristic application of the circle method suggests that for s ≥ 4, one should have the asymptotic formula (1.1) Rs(n) = Γ(4/3) Γ(s/3)Ss(n)n s/3−1 + o(ns/3−1), as n→∞, where

Additive Representation in Thin Sequences, VII: Restricted Moments of the Number of Representations