Let \ (A\) be a multiset with elements in an abelian group. Let \ (FS (A) \) be the multiset containing the \ (2^|A|\) sums of all subsets of \ (A\). We study the reconstruction problem "Given \ (FS (A) \), is it possible to identify \ (A\)? ". We prove that, up to identifying multisets through a natural equivalence relation, the function \ (A FS (A) \) is injective (and thus the reconstruction problem is solvable) if and only if every order \ (n\) of a torsion element of the abelian group satisfies a number-theoretical property related to the multiplicative group \ ( (Z/n Z) ^*\). The core of the proof relies on a delicate study of the structure of cyclotomic units. Moreover, as a tool, we develop an inversion formula for a novel discrete Radon transform on finite abelian groups that might be of independent interest. Mathematics Subject Classifications: 11P70, 05B10, 11R18, 44A12Keywords: Subset sums, inverse problems, Radon transform, cyclotomic extension
Ciprietti et al. (Fri,) studied this question.