# Small-sum pairs in abelian groups

Reza Akhtar^{[1]}; Paul Larson^{[1]}

- [1] Department of Mathematics Miami University Oxford, OH 45056, USA

Journal de Théorie des Nombres de Bordeaux (2010)

- Volume: 22, Issue: 3, page 525-535
- ISSN: 1246-7405

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topAkhtar, Reza, and Larson, Paul. "Small-sum pairs in abelian groups." Journal de Théorie des Nombres de Bordeaux 22.3 (2010): 525-535. <http://eudml.org/doc/116418>.

@article{Akhtar2010,

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 $a \in A$ 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.},

affiliation = {Department of Mathematics Miami University Oxford, OH 45056, USA; Department of Mathematics Miami University Oxford, OH 45056, USA},

author = {Akhtar, Reza, Larson, Paul},

journal = {Journal de Théorie des Nombres de Bordeaux},

keywords = {small-sum pair; quasi-periodic decomposition; Kemperman pair; aperiodic; Kneser theorem},

language = {eng},

number = {3},

pages = {525-535},

publisher = {Université Bordeaux 1},

title = {Small-sum pairs in abelian groups},

url = {http://eudml.org/doc/116418},

volume = {22},

year = {2010},

}

TY - JOUR

AU - Akhtar, Reza

AU - Larson, Paul

TI - Small-sum pairs in abelian groups

JO - Journal de Théorie des Nombres de Bordeaux

PY - 2010

PB - Université Bordeaux 1

VL - 22

IS - 3

SP - 525

EP - 535

AB - 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.

LA - eng

KW - small-sum pair; quasi-periodic decomposition; Kemperman pair; aperiodic; Kneser theorem

UR - http://eudml.org/doc/116418

ER -

## References

top- 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. Zbl1138.11044MR2289427
- D. Grynkiewicz, Quasi-periodic decompositions and the Kemperman structure theorem. European Journal of Combinatorics 26 (2005), 559–575. Zbl1116.11081MR2126639
- Y. O. Hamidoune, Subsets with small sums in abelian groups. I. The Vosper property. European Journal of Combinatorics 18 (1997), no. 5, 541–556. Zbl0883.05065MR1455186
- Y. O. Hamidoune, Subsets with a small sum. II. The critical pair problem. European Journal of Combinatorics 21 (2000), no. 2, 231–239. Zbl0941.05064MR1742437
- J. H. B. Kemperman, On small subsets of an abelian group. Acta Mathematica 103 (1960), 63–88. Zbl0108.25704MR110747
- 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. Zbl0072.03402MR77555
- M. Nathanson, Additive Number Theory. Springer, 1996. Zbl0859.11002MR1477155

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