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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