Small-sum pairs in abelian groups

Akhtar, Reza; Larson, Paul

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 $a\in A$ and $b\in B$ 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 $\left|A\right|=\left|B\right|=k$ et $|A+A|=|A+B|=2k-1$. Pour tous sous-ensembles $X$, $Y$ de $G$ et $c\in G$, notons ${\nu }_c(X,Y)$ le nombre de couples $(x,y)\in X\times Y$ tels que $c=x+y$. Nous résolvons une question de Bihani et Jin en montrant qu’il existe $g\in G$ tel que $A=g+B$ si $A+B$ est apériodique ou s’il existe $a\in A$ et $b\in B$ tels que ${\nu }_{a+b}(A,B)={\nu }_{a+a}(A,A)=1$. Nous donnons aussi une description explicite des divers contre-exemples qui se présentent si aucune de ces hypothèses n’est satisfaite.

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