Let $A$ be a multiset with elements in an abelian group. Let $FS(A)$ be its subset sums multiset, i.e., the multiset containing the $2^|A|$ sums of all subsets of $A$.
Given $FS(A)$, can you determine $A$?
If the abelian group is $\Bbb Z$, one can see that the two multisets $A = \{−2, 1, 1\}$ and $A' = \{−1, −1, 2\}$ satisfy $FS(A) = FS(A')$; notice that one is obtained from the other by changing signs to the elements. We will see that this is the only obstruction and so, up to the sign of the elements, $FS(A)$ determines $A$ in $\Bbb Z$.
In a general abelian group the situation is much more involved and we will see that the answer depends intimately on the orders of the torsion elements of the group. The core of the proof relies on a delicate study of the structure of cyclotomic units and on an inversion formula for a novel discrete Radon transform on finite abelian groups.
This is a joint work with Andrea Ciprietti.
Federico Glaudo