Small-sum pairs in abelian groups
Reza Akhtar1; Paul Larson1
1 Department of Mathematics Miami University Oxford, OH 45056, USA
Journal de théorie des nombres de Bordeaux, Volume 22 (2010) no. 3, pp. 525-535

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.

Soient G un groupe abélien fini et A, B deux sous-ensembles de G tels que |A|=|B|=k et |A+A|=|A+B|=2k-1. Pour tous sous-ensembles X, Y de G et c∈G, notons ν c (X,Y) le nombre de couples (x,y)∈X×Y tels que c=x+y. Nous résolvons une question de Bihani et Jin en montrant qu’il existe g∈G tel que A=g+B si A+B est apériodique ou s’il existe a∈A et b∈B tels que ν a+b (A,B)=ν a+a (A,A)=1. Nous donnons aussi une description explicite des divers contre-exemples qui se présentent si aucune de ces hypothèses n’est satisfaite.

DOI: 10.5802/jtnb.730

Reza Akhtar 1; Paul Larson 1

1 Department of Mathematics Miami University Oxford, OH 45056, USA 
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Reza Akhtar; Paul Larson. Small-sum pairs in abelian groups. Journal de théorie des nombres de Bordeaux, Volume 22 (2010) no. 3, pp. 525-535. doi: 10.5802/jtnb.730

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