Structure in sets with logarithmic doubling

TL;DR

The paper proves that in abelian groups, sets with small doubling contain large structured subsets, and applies this to generalize a theorem related to arithmetic progressions.

Contribution

It introduces a new structural result for sets with small doubling in abelian groups and applies it to extend the Roth-Meshulam theorem.

Findings

Existence of a large structured subset within sets with small doubling

Bound on the size of the structured subset in terms of K and log |A|

Application to a generalized Roth-Meshulam theorem

Abstract

Suppose that G is an abelian group, A is a finite subset of G with |A+A|< K|A| and eta in (0,1] is a parameter. Our main result is that there is a set L such that |A cap Span(L)| > K^{-O_eta(1)}|A| and |L| = O(K^eta log |A|). We include an application of this result to a generalisation of the Roth-Meshulam theorem due to Liu and Spencer.