Small-sum pairs in abelian groups
Journal de théorie des nombres de Bordeaux, Tome 22 (2010) no. 3, pp. 525-535.

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 cG, 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 gG tel que A=g+B si A+B est apériodique ou s’il existe aA et bB 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.

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 aA and bB 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.

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, Tome 22 (2010) no. 3, pp. 525-535. doi : 10.5802/jtnb.730. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.730/

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