Unique sums and differences in finite Abelian groups

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 o...

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Main Authors: Leung, Ka Hin, Schmidt, Bernhard
其他作者: School of Physical and Mathematical Sciences
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語言:English
出版: 2022
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spelling sg-ntu-dr.10356-1597692022-07-01T07:32:21Z Unique sums and differences in finite Abelian groups Leung, Ka Hin Schmidt, Bernhard School of Physical and Mathematical Sciences Science::Mathematics Finite Abelian Groups Sumsets 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. Ministry of Education (MOE) This research is supported by the Ministry of Education, Singapore, under its Academic Research Fund Tier 1 (RG27/18). 2022-07-01T07:32:21Z 2022-07-01T07:32:21Z 2022 Journal Article Leung, K. H. & Schmidt, B. (2022). Unique sums and differences in finite Abelian groups. Journal of Number Theory, 233, 370-388. https://dx.doi.org/10.1016/j.jnt.2021.06.014 0022-314X https://hdl.handle.net/10356/159769 10.1016/j.jnt.2021.06.014 2-s2.0-85115767610 233 370 388 en RG27/18 Journal of Number Theory © 2021 Elsevier Inc. All rights reserved.
institution Nanyang Technological University
building NTU Library
continent Asia
country Singapore
Singapore
content_provider NTU Library
collection DR-NTU
language English
topic Science::Mathematics
Finite Abelian Groups
Sumsets
spellingShingle Science::Mathematics
Finite Abelian Groups
Sumsets
Leung, Ka Hin
Schmidt, Bernhard
Unique sums and differences in finite Abelian groups
description 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.
author2 School of Physical and Mathematical Sciences
author_facet School of Physical and Mathematical Sciences
Leung, Ka Hin
Schmidt, Bernhard
format Article
author Leung, Ka Hin
Schmidt, Bernhard
author_sort Leung, Ka Hin
title Unique sums and differences in finite Abelian groups
title_short Unique sums and differences in finite Abelian groups
title_full Unique sums and differences in finite Abelian groups
title_fullStr Unique sums and differences in finite Abelian groups
title_full_unstemmed Unique sums and differences in finite Abelian groups
title_sort unique sums and differences in finite abelian groups
publishDate 2022
url https://hdl.handle.net/10356/159769
_version_ 1738844826085883904