Decompositions, approximate structure, transference, and the Hahn-Banach theorem

TL;DR

This paper surveys additive combinatorics results and explores the utility of the finite-dimensional Hahn-Banach theorem, providing simplified proofs of key theorems including the Green-Tao theorem and its transference principle.

Contribution

It offers simplified proofs of important results in additive combinatorics, highlighting the usefulness of the Hahn-Banach theorem in this field.

Findings

Simplified proof of a key step in the Green-Tao theorem

Application of Hahn-Banach theorem to additive combinatorics

Independent simplified proof of Green-Tao transference principle

Abstract

This paper is partly a survey of certain kinds of results and proofs in additive combinatorics, and partly a discussion of how useful the finite-dimensional Hahn-Banach theorem can be. The most interesting single result is probably a simpler proof of a key step in the proof of the Green-Tao theorem, but several other applications of the method are given. A similarly simplified proof of the Green-Tao transference principle was obtained independently (and expressed in a rather different language) by Reingold, Trevisan, Tulsiani and Vadhan.