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)?

primecongruent to 1 mod 4 is the sum of two squares. But the primes aren't dense in Z. :) Post edited. Actually for my application densities like c*x/poly(log x) would be sufficient...but no need to complicate things. – David Mandell Freeman Jan 22 '10 at 1:56