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- W4210803017 abstract "Let A,B be subsets of a finite abelian group G. Suppose that A+B does not contain a unique sum, i.e., there is no g∈G with a unique representation g=a+b, a∈A, b∈B. From such sets A,B, sparse linear systems over the rational numbers arise. We obtain a new determinant bound on invertible submatrices of the coefficient matrices of these linear systems. Under the condition that |A|+|B| is small compared to the order of G, these bounds provide essential information on the Smith Normal Form of these coefficient matrices. We use this information to prove that A and B admit coset partitions whose parts have properties resembling those of A and B. As a consequence, we improve previously known sufficient conditions for the existence of unique sums in A+B and show how our structural results can be used to classify sets A and B for which A+B does not contain a unique sum when |A|+|B| is relatively small. Our method also can be applied to subsets of abelian groups which have no unique differences." @default.
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- W4210803017 date "2021-02-01" @default.
- W4210803017 modified "2023-09-25" @default.
- W4210803017 title "Abstracts" @default.
- W4210803017 doi "https://doi.org/10.1016/j.hm.2020.10.002" @default.
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