# Small-sum pairs in abelian groups

Reza Akhtar[1]; Paul Larson[1]

• [1] Department of Mathematics Miami University Oxford, OH 45056, USA
• Volume: 22, Issue: 3, page 525-535
• ISSN: 1246-7405

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## Abstract

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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.

## How to cite

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Akhtar, 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

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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. Zbl1138.11044MR2289427
2. D. Grynkiewicz, Quasi-periodic decompositions and the Kemperman structure theorem. European Journal of Combinatorics 26 (2005), 559–575. Zbl1116.11081MR2126639
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. Zbl0883.05065MR1455186
4. 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
5. J. H. B. Kemperman, On small subsets of an abelian group. Acta Mathematica 103 (1960), 63–88. Zbl0108.25704MR110747
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. Zbl0072.03402MR77555
7. M. Nathanson, Additive Number Theory. Springer, 1996. Zbl0859.11002MR1477155

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