On the determination of sets by their subset sums

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