# What is the best known upper bound for G(k) in Waring's Problem?

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?

• The breakthroughs on Vinogradov's mean value theorem don't seem to give improvements on $G(k)$. The best results here remain Wooley's work from the 90's giving results of the form $G(k) \le (1+o(1)) k\log k$ (and refinements on making the $o(1)$ here explicit). – Lucia Nov 2 '19 at 16:24
• @Lucia Thank you! – DaveT Nov 2 '19 at 17:50

While not exactly the classical quantity $$G(k)$$, there are partial improvements on asymptotic behavior for the Waring-Goldbach problem, namely for $$H(k)$$, defined to be the least integer $$s$$ such that every sufficiently large positive integer congruent to $$s$$ modulo $$K(k)$$ may be written as $$p_1^k + p^k_2 + ··· + p^k _s = n,$$ where $$p_1,\dots,p_s$$ are prime numbers and where $$K(k)=\Pi_{(p-1)|k} p^{\gamma}$$, and $$\gamma$$ is a parameter defined below. Shortly after Bourgain, Demeter, and Guth's resolving of the Vinogradov's conjecture Kumcev and Wooley used their results to improve Hua's estimate
$$H(k)\leq k(4 \log k + 2 \log \log k + O(1)) \text{ as }k\rightarrow\infty, \text{ to }$$
$$H(k)\leq (4k − 2) \log k + k − 4 \text{ for } k\geq 3.$$
$$H(k) ≤ (4k − 2)\log k − (2 \log 2 − 1)k − 3$$
holding for large $$k$$. To compute the value $$K(k)$$, let natural $$k$$ and prime number $$p$$ be given, and define $$\theta = \theta(k, p)$$ to be the integer with $$p^\theta|k$$ but $$p^{\theta+1}\nmid k$$, and $$\gamma=\gamma(k, p)$$ by $$γ(k, p) =\begin{cases} \theta + 2,\ \text{ when } p = 2 \text{ and }\theta > 0,\\ \theta + 1, \text{ otherwise.} \end{cases}$$