A Freiman-type theorem for locally compact abelian groups

TL;DR

This paper extends Freiman's theorem to locally compact abelian groups, characterizing subsets with small sumsets under certain measure growth conditions, up to logarithmic factors.

Contribution

It provides a new Freiman-type structural theorem for subsets of locally compact abelian groups with controlled measure growth.

Findings

Characterization of subsets with small sumsets in locally compact abelian groups.

Description of such sets up to logarithmic factors in the dimension d.

Extension of Freiman's theorem beyond discrete groups.

Abstract

We prove a Freiman-type theorem for locally compact abelian groups. If A is a subset of a locally compact abelian group with Haar measure m and m(nA) < n^dm(A) for all n>d log d then we describe A in a way which is tight up to logarithmic factors in d.