On The Determination of Sets By Their Subset Sums

TL;DR

This paper investigates whether a multiset's original elements can be reconstructed solely from the sums of all its subsets within abelian groups, providing a complete characterization for when this is possible.

Contribution

It establishes a necessary and sufficient condition for the injectivity of the subset sum function in abelian groups, linking it to number-theoretical properties of torsion element orders.

Findings

Reconstruction is possible if and only if torsion element orders satisfy specific number-theoretic conditions.

Develops an inversion formula for a new discrete Radon transform on finite abelian groups.

Provides a comprehensive analysis of subset sum multisets in abelian groups.

Abstract

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 $A$?'', and we give a satisfactory answer for all abelian groups. We prove that, up to identifying multisets through a natural equivalence relation, the function $A \mapsto FS(A)$ 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 certain number-theoretical property linked 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.