A structure theorem for sets with doubling $4+\delta$

TL;DR

This paper establishes a structural theorem for integer sets with small doubling constants slightly above 4, extending previous results and advancing understanding in additive combinatorics.

Contribution

It generalizes earlier work on sets with doubling less than 4 to include those with doubling up to 4 + δ, for small δ, addressing a question posed by Green.

Findings

Proves a structural result for sets with doubling at most 4 + δ

Extends previous results from doubling less than 4 to slightly above 4

Progresses towards answering Green's open question

Abstract

We prove a structural result for sets of integers with doubling at most $4 + \delta$, with $\delta>0$ sufficiently small. This generalises earlier work of Eberhard--Green--Manners which dealt with sets of integers with doubling strictly less than $4$, and makes progress towards a question of Green.