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.

Reçu le :
Publié le :
DOI : https://doi.org/10.5802/jtnb.730
@article{JTNB_2010__22_3_525_0,
     author = {Reza Akhtar and Paul Larson},
     title = {Small-sum pairs in abelian groups},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {525--535},
     publisher = {Universit\'e Bordeaux 1},
     volume = {22},
     number = {3},
     year = {2010},
     doi = {10.5802/jtnb.730},
     zbl = {1236.11026},
     mrnumber = {2769329},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.730/}
}
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/

[1] P. Bihani and R. Jin, Kneser’s Theorem for Upper Banach Density. Journal de Théorie des Nombres de Bordeaux 18 (2006), no. 2, 323–343. | Numdam | MR 2289427 | Zbl 1138.11044

[2] D. Grynkiewicz, Quasi-periodic decompositions and the Kemperman structure theorem. European Journal of Combinatorics 26 (2005), 559–575. | MR 2126639 | Zbl 1116.11081

[3] Y. O. Hamidoune, Subsets with small sums in abelian groups. I. The Vosper property. European Journal of Combinatorics 18 (1997), no. 5, 541–556. | MR 1455186 | Zbl 0883.05065

[4] Y. O. Hamidoune, Subsets with a small sum. II. The critical pair problem. European Journal of Combinatorics 21 (2000), no. 2, 231–239. | MR 1742437 | Zbl 0941.05064

[5] J. H. B. Kemperman, On small subsets of an abelian group. Acta Mathematica 103 (1960), 63–88. | MR 110747 | Zbl 0108.25704

[6] A. G. Vosper, The critical pairs of subsets of a group of prime order. J. London Math. Soc. 31 (1956), 200–205 and 280–282. | MR 77555 | Zbl 0072.03402

[7] M. Nathanson, Additive Number Theory. Springer, 1996. | MR 1477155 | Zbl 0859.11002

Cité par document(s). Sources :