 
Summary: Smallsum pairs in abelian groups
Reza Akhtar
Dept. of Mathematics
Miami University, Oxford, OH 45056, USA
reza@calico.mth.muohio.edu
Paul Larson
Dept. of Mathematics
Miami University, Oxford, OH 45056, USA
larsonpb@muohio.edu
December 29, 2009
Abstract
Let G be an abelian group and A, B two subsets of equal size k such that
A + B and A + A both have size 2k  1. Answering a question of Bihani and
Jin, we prove that if A + B is aperiodic or if there exist elements a # A and
b # B such that a+ b has a unique expression as an element of A+B and a+a
has a unique expression as an element of A + A, then A is a translate of B.
We also give an explicit description of the various counterexamples which arise
when neither condition holds.
1 Introduction
Let G be an abelian group, written additively. If A and B are subsets of G, we write
