I have a number theory situation that I hope someone will recognize as a known situation and can direct me to some relevant papers in the literature.
This came out of some numerical experiments run by myself and some research collaborators on the coefficients of certain eta quotients, modulo 4.
I had this question posted on Stack Exchange Mathematics, but they closed it there (did not like the way the question was formulated) - let me know if there other details I can supply.
Let $S_1$ be an infinite subset of $N_0 =\{0,1,2,3,...\}$.
Let $S_2 = N_0\setminus S_1$ (complement of $S_1$ in $N_0$, also infinite).
Let $S^*$ be a third set of positive integers, possibly sparse, such as $\{n^2|n\geq 1\}$ or $\{n^2|n\geq 1\}\cup \{2n^2|n\geq 1\}$.
Let $n_2$ be an arbitrary element of $S_2$.
It seems that for certain choices of $S_1$ (and thus $S_2$) and $S^*$ that the number of solutions to the equation $n_1+n^* = n_2, n_1 \in S_1, n^* \in S^*$, is always even.
As an example, take $S_1=\{n(3n+1)/2|n\in \mathbb{Z}\}$ and $S^*=\{n^2|n\geq 1\}\cup \{2n^2|n\geq 1\}$. So $S_1=\{\dots, 287, 247, 210, 176, 145, 117, 92, 70, 51, 35, 22, 12, 5, 1, 0, 2, 7, 15, 26, 40, 57, 77, 100, 126, 155, 187, 222, 260, 301, \dots \}$ and $S^*=\{1, 2, 4, 8, 9, 16, 18, 25, 32, 36, 49, 50, 64, 72, 81, 98, 100, 121, 128, 144, 162, 169, 196, 200, 225, 242, 256, 288,\dots \}$.
Consider $n_2=226\in S_2$. The list of solutions $(n_1,n^*)$ to $n_1+n^* = n_2, n_1 \in S_1, n^* \in S^*$ is $\{(222, 4), (210, 16), (145, 81), (126, 100), (57, 169), (1, 225), (176, 50), (26, 200) \}$.
It can be seen that he number of solutions is 8 (even), and this also appears to be the case for any other integer in $S_2$.
I am not asking for a proof for this particular case (one of my collaborators has done it). Instead, I am asking anyone knows of other papers about this phenomenon?
Does this situation ring any bells for anyone? If so, I would appreciate any pointers to relevant papers in the literature. This may be something well known in, say, additive number theory, but that is not an area that I know to any great extent.
Someone I was in grad school with pointed out this paper to me, which is about the topic I discussed above: https://arxiv.org/abs/math/0506496
