If we have $N$ sets, $\{{A}_{1},\dots ,{A}_{N}\}$, and we form a set $S$ by taking the sum of each element in the set with each element in the other sets, what can we say about the mode of $S$?

Intuitively, I would like to think that we can simply take the sum of the modes, i.e:

$\mathrm{Mode}(S)=\sum _{n=1}^{N}\mathrm{Mode}({A}_{n})$

However, this seems unlikely, especially as we would expect that $\mathrm{Mode}({A}_{n})$ could potentially be a set of values, rather than a single value.

So I was wondering if we'd be able to relax this condition to state that $\mathrm{Mode}(S)\subseteq \sum _{n=1}^{N}\mathrm{Mode}({A}_{n})$, where we define $\mathrm{Mode}(A)+\mathrm{Mode}(B)$ as the set formed by taking the sum of each element in $\mathrm{Mode}(A)$ with each element in $\mathrm{Mode}(B)$, formally:

$\mathrm{Mode}(S)\subseteq \sum _{n=1}^{N}\mathrm{Mode}({A}_{n})=\{\sum _{n=1}^{N}{x}_{n}:{x}_{i}\in {A}_{i}\}$

This seems to be true, but I was wondering if we could say anything stronger?