Unique sums and differences in finite Abelian groups

• Main Authors: Leung, Ka Hin, Schmidt, Bernhard
• Other Authors: School of Physical and Mathematical Sciences
• Format: Article
• Language: English
• Published: 2022
• Subjects: Science::Mathematics; Finite Abelian Groups; Sumsets
• Online Access: https://hdl.handle.net/10356/159769
• Institution: Nanyang Technological University

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.