Sum-free sets and arithmetic notions of structure in combinatorics

Thesis

This thesis is concerned with four problems in additive combinatorics, each of which studies a certain notion of arithmetic structure and which properties of a general additive set, such as its density or the structure of the group in which it is contained, imply that it necessarily exhibits such structure. One of the main results of this thesis shows that any set A of N positive integers contains a subset A′ ⊂ A of size |A′| ⩾ ^N/3+ c log log N which is sum-free, meaning that there are no three x, y, z ∈ A′ with x + y = z. This answers a longstanding problem of Erdos from 1965.

Authors: Bedert, B

• DOI: 10.5287/ora-rdq7ybpdz
• Type of award: DPhil
• Awarding institution: University of Oxford
• Language: English
• Keywords: Fourier analysis; additive combinatorics; sum-free sets