Davenport's method and slim exceptional sets: the asymptotic formulae in Waring's problem

We apply a method of Davenport to improve several estimates for slim exceptional sets associated with the asymptotic formula in Waring’s problem. In particular, we show that the anticipated asymptotic formula in Waring’s problem for sums of seven cubes holds for all but $O(N^{1\mathrm{/}3+ϵ})$ of the natural numbers not exceeding $N$.