Unique sums and differences in finite Abelian groups

Abstract

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.