Combinatorial Theory

Let \(A\) be a multiset with elements in an abelian group. Let \(\operatorname{FS}(A)\) be the multiset containing the \(2^{|A|}\) sums of all subsets of \(A\). We study the reconstruction problem "Given \(\operatorname{FS}(A)\), is it possible to identify \(A\)?". We prove that, up to identifying multisets through a natural equivalence relation, the function \(A \mapsto \operatorname{FS}(A)\) 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 \((\mathbb{Z}/n \mathbb{Z})^*\). 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, 44A12

Keywords: Subset sums, inverse problems, Radon transform, cyclotomic extension