Please use this identifier to cite or link to this item:
Title: Unique sums and differences in finite Abelian groups
Authors: Leung, Ka Hin
Schmidt, Bernhard
Keywords: Science::Mathematics
Issue Date: 2022
Source: Leung, K. H. & Schmidt, B. (2022). Unique sums and differences in finite Abelian groups. Journal of Number Theory, 233, 370-388.
Project: RG27/18
Journal: Journal of Number Theory
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.
ISSN: 0022-314X
DOI: 10.1016/j.jnt.2021.06.014
Schools: School of Physical and Mathematical Sciences 
Rights: © 2021 Elsevier Inc. All rights reserved.
Fulltext Permission: none
Fulltext Availability: No Fulltext
Appears in Collections:SPMS Journal Articles

Citations 50

Updated on Sep 22, 2023

Web of ScienceTM
Citations 50

Updated on Sep 22, 2023

Page view(s)

Updated on Sep 23, 2023

Google ScholarTM




Items in DR-NTU are protected by copyright, with all rights reserved, unless otherwise indicated.