TITLE: Variants of Waring's problem
QUESTION [13 upvotes]: Waring's problem (previously asked about here) asks, for each integer $k \ge 2$, what is the smallest integer $g(k)$ such that any positive integer can be written as a sum of $g(k)$ kth powers.  We have $g(2) = 4$, $g(3) = 9$, etc.  A (harder) variant asks what the smallest integer $G(k)$ is such that all sufficiently large integers can be written as a sum of $G(k)$ kth powers.  
I have two related questions:

What is known if we relax the condition ``any positive integer'' and only require a positive-density subset?  More precisely, we look for the smallest $g'(k)$ for which there is some $S \subset \mathbb{Z}_{>0}$ of positive density such that any $x \in S$ can be written as $g'(k)$ $k$th powers.  Then we have $g'(2) = 3$, while $G(2) = 4$; and $g'(3) = 4$, while it is only known that $4 \le G(3) \le 7$.  Is anything known about $g'(k)$ for k = 4,5, or larger?
For fixed k, is there an efficient algorithm that, given n, writes n as a sum of $g(k)$ kth powers?  What about decomposing n into the minimal number of kth powers for that n?   (Here `efficient' means polynomial in log(n).)  

Edit: Wikipedia says that ``In the absence of congruence restrictions, a density argument suggests that G(k) should equal k + 1.''  So perhaps this is the answer to (1)?

REPLY [15 votes]: Regarding question 1, it is easy to prove that $g'(k) \ge k$, and the expectation is that $g'(k)=k$ for all $k \ge 3$.  But the equality is an open question even for $k=3$.  See Deshouillers, Jean-Marc; Hennecart, François; Landreau, Bernard; Sums of powers: an arithmetic refinement to the probabilistic model of Erdös and Rényi.  Acta Arith. 85 (1998), no. 1, 13-33, and other papers by these authors.