Arithmetic progressions in sets of small doubling

TL;DR

This paper proves that large subsets of abelian groups with small doubling contain non-trivial three-term arithmetic progressions and their sumsets contain long arithmetic progressions or subgroup cosets, extending previous results.

Contribution

It extends known results on arithmetic progressions in small doubling sets from integers and finite fields to arbitrary abelian groups.

Findings

Sets with small doubling contain non-trivial 3-term arithmetic progressions.

Sumsets of such sets contain long arithmetic progressions or subgroup cosets.

Results generalize previous work from specific groups to all abelian groups.

Abstract

We show that if a finite, large enough subset A of an arbitrary abelian group satisfies the small doubling condition |A + A| < (log |A|)^{1 - epsilon} |A|, then A must contain a three-term arithmetic progression whose terms are not all equal, and A + A must contain an arithmetic progression or a coset of a subgroup, either of which of size at least exp^[ c (log |A|)^{delta} ]. This extends analogous results obtained by Sanders and, respectively, by Croot, Laba and Sisask in the case where the group is that of the integers or a finite field.