Sum set fixpoint, how many iterations?

Question

I want to approach linear equations of the following form over the integers $\mathbb{Z}$:

$$x_1 + \cdots + x_n = 0.$$

I stepped over the sum set, which is defined as follows:

$$S + T = \{ x + y \mid x \in S, y \in T \}.$$

Now assume we know that $x_i \in S_i^0$, where each $S_i^0$ is a finite integer set. One can define the following mapping which sends $(S_1^k,\dots,S_n^k)$ to $(S_1^{k+1},\dots,S_n^{k+1})$:

$$S_i^{k+1} = (- \sum_{j \neq i} S_j^k) \cap S_i^k.$$

How many iterations $l$ does it take until the mapping reaches a fixed point, i.e. what is the smallest $l$ such that $(S_1^l,\dots,S_n^l) = (S_1^{l+1},\dots,S_n^{l+1})$? Is there a more direct way to compute the fixed point?

Bye

Answer 1

The fixed point is reached after at most one iteration.

Indeed $S_i^1=\{x_i\in S_i^0: \exists(x_j)_{j\ne i}\in\prod_{j\ne i} S_j^0: x_1+\dots+x_n=0\}$.

Therefore all such $x_j$ will remain in $S_j^1$, so we have $(x_j)_{j\ne i}\in\prod_{j\ne i} S_j^1$ and therefore $S_i^2=S_i^1$.

Now of course I don't know if that's what you want: such a reduction won't give you a way to find a particular solution, unless you apply your method recursively. For example, you could have $S_i^1=S_i^0$ for all $i$, even though there might only be $|S_i^0|=k$ solutions among the $k^n$ candidates.

Answer 2

A trivial upper bound is $\sum\limits_{i=1}^n |S_i^0|$, since each $S_i^{k+1}\subseteq S_i^{k}$ and so each iteration which is not a fixed point reduces $\sum\limits_{i=1}^n |S_i^k|$ by at least $1$. For sufficiently large $n$ and large sets $S_i^0$ this bound may not be optimal.