On the Structure of Additive Systems of Integers

TL;DR

This paper explores the structure of additive systems of integers, establishing connections between sum systems, sum-and-distance systems, and reversible cuboids, and provides a construction method based on factorisations.

Contribution

It introduces a bijection between sum systems and sum-and-distance systems, and characterizes sum systems as principal reversible cuboids with an explicit construction formula.

Findings

Established a bijection between sum systems and sum-and-distance systems.

Proved sum systems are equivalent to principal reversible cuboids.

Provided an explicit construction formula for all sum systems.

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.