Let $S_1,\cdots,S_k$ be $k$ infinite sets of positive integers. Let $N_i(z)$ be the numbers of elements in $S_i$ that are less or equal to $z$. Let us further assume that $$N_i(S) \sim \frac{a_i z^{b_i}}{(\log z)^{c_i}} \mbox{ as } z\rightarrow \infty,$$ where $a_i>0, 0 < b_i \leq 1, c_i \geq 0 (i=1,\cdots k)$ are constants. Also, let
- $r(z)$ be the number of solutions to $x_1+\cdots + x_k\leq z$, with $x_i \in S_i$ (here $z$ is an integer).
- $t(z)$ be the number of solutions to $x_1+\cdots + x_k = z$, with $x_i \in S_i$ (here $z$ is an integer).
Note that $t(z) = r(z) - r(z-1) \sim r'(z)$ asymptotically on average, where $r'$ denotes the derivative.
Question: Is the following result correct?
$$r(z) \sim \frac{\prod_{i=1}^k a_i \Gamma(b_i+1)}{\Gamma(1+\sum_{i=1}^k b_i)} \cdot \frac{z^{b_1 + \cdots + b_k}}{(\log z)^{c_1 +\cdots + c_k}} \mbox{ as } z \rightarrow \infty. $$
This covers many cases such as
- Sums of squares
- Sums of primes
- Sums of powers
- Sums of a square and a prime
- Sums of superprimes
However it assumes that there are no congruence restrictions (for instance sums of odd integers can not be an odd integer, thus it does not work for sums of two odd primes, but it does for sums of two pseudo-primes depending on how you define pseudo-primes). See a proof for the case $k=2$, here. Also, is it true that if $r'(z) \rightarrow\infty$ as $z\rightarrow\infty$, then almost all integers can be written as $$z = x_1+\cdots + x_k, \mbox{ with } x_i \in S_i.$$ By almost, I mean all but a finite number, or maybe all but a small infinite subset of positive integers, of density zero. If this was true, we would have the following result, maybe not that hard to prove:
Almost all positive integers can be written as $x_1^{d_1} + \cdots + x_k^{d_k}$ if $\frac{1}{d_1} + \cdots + \frac{1}{d_k}> 1$.
Here the $d_i$'s and $x_i$'s are positive integers. The constants $d_1,\cdots,d_k$ are fixed. This is compatible with a conjecture associated with the Waring problem, in the absence of congruence restrictions (see Wikipedia entry, check out the section Lower Bounds for $G(k)$). Namely, the fact that almost all positive integers can be written as $x_1^k + x_2^k +\cdots + x_{k+1}^k$ in the absence of congruence restrictions (e.g. not true for $k=2, 4$ due to congruence restrictions, but potentially true for $k=3,5,7, 11, 13$ etc.)
Update on 7/9/2022:
The formula for $r(z)$ provides an asymptotic upper bound to the number of elements in $S=S_1+\cdots +S_k$ that are smaller than $z$ (here $k$ is fixed). The actual value can be lower by an order of magnitude: this is the case for sums of squares. See also the multivariate Euler-Maclaurin summation formula, here.
