bibkey: ciprietti2023subsetsums
authors: Andrea Ciprietti, Federico Glaudo
year: 2023
title: "On The Determination of Sets By Their Subset Sums"
doi: 10.48550/arXiv.2301.04635
url: https://arxiv.org/abs/2301.04635v2
claim: "An abelian group G is FS-regular if and only if ord(g) belongs to O_FS for every finite-order element g."
strata_touched:
  - D5/S3/Combinatorics/SubsetSpectrum/NonnegativeRecovery
license: citation-only
triage: anchor

Recovery from the full subset-sum multiset

The introduction defines the full subset-sum multiset with all indexed subset occurrences, and the equivalence relation generated by reversing a zero-sum submultiset. The main theorem, labelled thm:main in the source, characterizes the abelian groups for which these are precisely the fibers of the full-spectrum map. In particular, the additive group of the reals has this property, since its only finite-order element is zero, of order one, and one belongs to O_FS.

For nonnegative real coefficients the resulting recovery is exact: reversing a zero-sum submultiset can affect only zero coefficients. This known consequence is the supplier used in NonnegativeRecovery; no originality is claimed for it. The formal proof uses the least nonempty subset sum and cancellation in the natural-coefficient additive monoid algebra. Removing one numerical zero removes the empty-subset occurrence while retaining all other zero-sum occurrences.

The odd-length even-spectrum classification additionally uses the first moment, complement reflection, and a parity obstruction with zero-coordinate repair. Those parity-restricted conclusions are separate from this full-spectrum source.

Verified locator

The cited source is doi:10.48550/arXiv.2301.04635, Theorem 1.1 (the source’s thm:main). Its zero-sum sign equivalence scope is the equivalence generated by reversing zero-sum submultisets; this note uses that scope for the full-subset-sum recovery statement and its nonnegative-real consequence.