Reconstructing a set from its subset sums: $2$-torsion-free groups
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866915053771423744 |
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| author | Glaudo, Federico Kravitz, Noah |
| author_facet | Glaudo, Federico Kravitz, Noah |
| contents | For a finite multiset $A$ of an abelian group $G$, let $\text{FS}(A)$ denote the multiset of the $2^{|A|}$ subset sums of $A$. It is natural to ask to what extent $A$ can be reconstructed from $\text{FS}(A)$. We fully solve this problem for $2$-torsion-free groups $G$ by giving characterizations, both algebraic and combinatorial, of the fibers of $\text{FS}$. Equivalently, we characterize all pairs of multisets $A,B$ with $\text{FS}(A)=\text{FS}(B)$. Our results build on recent work of Ciprietti and the first author. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2305_11062 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Reconstructing a set from its subset sums: $2$-torsion-free groups Glaudo, Federico Kravitz, Noah Combinatorics Number Theory For a finite multiset $A$ of an abelian group $G$, let $\text{FS}(A)$ denote the multiset of the $2^{|A|}$ subset sums of $A$. It is natural to ask to what extent $A$ can be reconstructed from $\text{FS}(A)$. We fully solve this problem for $2$-torsion-free groups $G$ by giving characterizations, both algebraic and combinatorial, of the fibers of $\text{FS}$. Equivalently, we characterize all pairs of multisets $A,B$ with $\text{FS}(A)=\text{FS}(B)$. Our results build on recent work of Ciprietti and the first author. |
| title | Reconstructing a set from its subset sums: $2$-torsion-free groups |
| topic | Combinatorics Number Theory |
| url | https://arxiv.org/abs/2305.11062 |