On the structure of additive systems of integers

Huxley, M. N., Lettington, M. C. ORCID: https://orcid.org/0000-0001-9327-143X and Schmidt, K. M. ORCID: https://orcid.org/0000-0002-0227-3024 2019. On the structure of additive systems of integers. Periodica Mathematica Hungarica 78 , pp. 178-199. 10.1007/s10998-018-00275-w

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Abstract

A sum-and-distance system is a collection of finite sets of integers such that the sums and differences formed by taking one element from each set generate a prescribed arithmetic progression. Such systems, with two component sets, arise naturally in the study of matrices with symmetry properties and consecutive integer entries. Sum systems are an analogous concept where only sums of elements are considered. We establish a bijection between sum systems and sum-and-distance systems of corresponding size, and show that sum systems are equivalent to principal reversible cuboids, which are tensors with integer entries and a symmetry of ‘reversible square’ type. We prove a structure theorem for principal reversible cuboids, which gives rise to an explicit construction formula for all sum systems in terms of joint ordered factorisations of their component set cardinalities.

Item Type:	Article
Date Type:	Publication
Status:	Published
Schools:	Mathematics
Additional Information:	This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Publisher:	Springer Verlag / Akadémiai Kiadó
ISSN:	0031-5303
Date of First Compliant Deposit:	22 November 2018
Date of Acceptance:	22 November 2018
Last Modified:	05 May 2023 00:49
URI:	https://orca.cardiff.ac.uk/id/eprint/117030

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