Arithmetic progressions in sets of small doubling

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.

30 pages, improved exposition

References in corpus (6)

• On the Bogolyubov-Ruzsa lemma
• On Roth's theorem on progressions
• The structure theory of set addition revisited
• New Proofs of Plünnecke-type Estimates for Product Sets in Groups
• On certain other sets of integers
• Green's sumset problem at density one half

Cited by in corpus (1)

• Complete type amalgamation for non-standard finite groups