**Waring's problem**: for fixed $k, s$ write natural numbers $n$ in the form $n=x_1^k+\dots+x_s^k$.
$$
G(k):= \min\{s:\text{ all sufficiently large $n$ can be written as above}\}
$$
A quick search / Wikipedia / Vaughan-Wooley's survey paper all suggest that the best upper bound for $G(k)$ is (Wooley)
$$
G(k)\leq k\log k+k\log\log k+O(k).
$$

Recent work of Bourgain, Demeter, Guth (https://arxiv.org/abs/1512.01565) and Wooley (https://arxiv.org/abs/1708.01220) on Vinogradov's theorem seems to improve several quantities related to Waring's problem and similar questions (eg. $g(k)$, which is as $G(k)$ but for all $n$). But I can't find any evidence of improvements in $G(k)$ following from this result. Does anyone know if such improvements are possible, and if so where to find them?