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On topics related to sum systems

Law, Ambrose 2023. On topics related to sum systems. PhD Thesis, Cardiff University.
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Abstract

For m ∈ N, we say that the m integer sets A1, . . . , Am ⊂ N0, form an m-part sum system if their sumset is the target set Xm j=1 Aj = n a1 + · · · + am : aj ∈ Aj , j ∈ {1, . . . , m} o = { 0, 1, 2, . . . ,Ym j=1 |Aj | − 1 } . That is to say, the sum over each element of the sets A1, . . . , Am uniquely generates the consecutive integers from 0 to Qm j=1 |Aj | − 1 with each integer appearing exactly once. Huxley, Lettington and Schmidt, in 2018, established a bijection between sum systems and sum-and-distance systems, utilising joint ordered factorisations, a specific form of ordered multi-factorisations, historically considered by MacMahon. They proved that for each m-part sum system there exists a corresponding m-part sum-and-distance system which generates the centro-symmetric set of consecutive (half) integers symmetric around the origin { − 1/2 (Ym j=1 |Aj | − 1 ), . . . , 1/2 (Ym j=1 |Aj | − 1 ) . In this thesis, we extend the results of Huxley, Lettington and Schmidt to obtain a unifying theory underpinning sum-and-distance systems, expressing their structures in terms of joint ordered-factorisations, thus enabling explicit construction formulae to be established via these factorisations. This unifying theory occurs when one allows consecutive half integers in the target set, when at least one component sum-and-distance set has even cardinality, leading to an invariance in the sum over weighted averages of the sum of squares across the sum-and-distance system component sets to be deduced. Further results include the application of associated divisor functions and Stirling numbers of the second kind, to enumerate all m-part joint ordered factorisations Nm(N) for a given positive integer N = n1 × n2 × . . . nm. We go on to show that the counting function Nm(N) satisfies an implicit three term recurrence relation proving an important relation in additive combinatorics. Additionally, sum systems (mod N + z), are considered, as well as orbit structures arising from very simple joint ordered factorisations. The latter leads to connections with cyclotomy.

Item Type: Thesis (PhD)
Date Type: Completion
Status: Unpublished
Schools: Mathematics
Subjects: Q Science > QA Mathematics
Funders: EPSRC
Date of First Compliant Deposit: 24 July 2024
Last Modified: 26 Jul 2024 08:38
URI: https://orca.cardiff.ac.uk/id/eprint/170885

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